Color-corrected lens systems

ABSTRACT

A method is described for selecting optical materials to use in designing color-corrected optical systems. Various dioptric, catadioptric and anamorphic optical systems are described, which use only two different types of optical materials to obtain precise axial color correction at three, four or five wavelengths (depending upon the particular optical materials), with only very small chromatic aberration occurring at wavelengths between the precisely color-corrected wavelengths. Examples of color-corrected lens systems comprising more than two glasses are also described.

This application is a division of Ser. No. 419,705 filed on Sept. 20,1982.

TECHNICAL FIELD

This invention relates generally to color-corrected optical systems.More particularly, this invention pertains to the selection of opticalmaterials for designing optical systems that are color-corrected at morethan two wavelengths.

BACKGROUND ART

The optical region of the electromagnetic spectrum is generallyunderstood to extend from the far ultraviolet wavelengths (about 0.36μm) to the near infrared wavelengths (about 1.10 μm). The selection ofoptical glasses for designing lens systems that are color-corrected atmore than two wavelengths in the optical region has been discussed by anumber of investigators, including:

(a) N.v.d.W. Lessing, J. Opt. Soc. Am. 47, 955 (1957)

(b) N.v.d.W. Lessing, J. Opt. Soc. Am. 48, 269 (1958)

(c) N.v.d.W. Lessing, Appl. Opt. 9, 1665 (1970)

(d) R. E. Stephens, J. Opt. Soc. Am. 49, 398 (1959)

(e) R. E. Stephens, J. Opt. Soc. Am. 50, 1016 (1960)

(f) M. Herzberger, Optica Acta 6, 197 (1959)

(g) M. Herzberger and N. R. McClure, Appl. Opt. 2, 553 (1963)

(h) M. Herzberger, Optik 35, 1 (1972)

(i) H. Drucks, Optik 23, 523 (1966)

(j) H. Schulz, Optik 25, 203 (1967)

(k) H. Schulz, Optik 25, 208 (1967)

(l) H. Pulvermacher, Optik 30, 297 (1969)

(m) H. Pulvermacher, Optik 30, 396 (1970)

(n) M. Herzberger and H. Pulvermacher, Optica Acta 17, 349 (1970)

(o) B. L. Nefedov, Sov. J. Opt. Technol. 40, 46 (1973)

(p) A. B. Agurok, Sov. J. Opt. Technol. 44, 114 (1977)

(q) M. G. Shpyakin, Sov. J. Opt. Technol. 45, 81 (1978)

(r) M. G. Shpyakin, Sov. J. Opt. Technol. 45, 219 (1978)

(s) G. A. Mozharov, Sov. J. Opt. Technol. 44, 146 (1977)

(t) G. A. Mozharov, Sov. J. Opt. Technol. 47, 398 (1980).

An optical "multiplet" is an optical system comprising a number(designated generally herein by k) of refracting elements made of anumber (designated generally herein by q) of different opticalmaterials, where each refracting element is made entirely of aparticular optical material, with the various refracting elements incombination providing color correction at a number (designated generallyherein by n) of wavelengths. The refracting elements could be configuredas lens elements, prism elements, or anamorphic elements, depending uponthe nature of the optical system. The term "color correction" as usedherein with respect to dioptric and catadioptric systems meanscorrection of axial chromatic aberration. Thus, to say that a lenssystem or a Mangin mirror system is color-corrected at n wavelengthsmeans that n wavelengths are brought to a common focus on the optic axisof the system. For a prism system, "color correction" means correctionof chromatic dispersion from the total deviation angle of the system.

Various combinations of optical glasses were discovered in the prior artfor designing lens systems that bring more than two wavelengths to acommon focus. Previous lens systems, which were color-corrected at morethan two wavelengths, generally required three or more different typesof optical glasses for making three or more lens elements. There waslittle success in the prior art in identifying suitable pairs of opticalglasses for designing lens doublets (i.e., lens systems comprising onlytwo lens elements) capable of bringing more than two wavelengths to acommon focus.

U.S. Pat. No. 3,395,962 to Herzberger, et al. described a photographicobjective comprising three lens triplets, with each triplet beingindividually color-corrected at four wavelengths. M. Herzberger, inreference (f) on the above list, suggested that color correction of anoptical lens system at four wavelengths is substantially equivalent tocolor correction for the entire spectrum bounded by the most widelyseparated of those four wavelengths.

A lens system comprising only two lens elements, i.e., a doublet, isinherently simpler than a lens system comprising a triplet orhigher-number multiplet of lens elements. In terms of fabrication costand compatibility with constraints imposed by the practicalconsiderations involved in designing optical instruments, a lens doubletthat is color-corrected at three or more wavelengths would ordinarily bepreferable to a higher-number lens multiplet color-corrected at the samenumber of wavelengths. A computer-aided search to identify pairs ofoptical glasses suitable for designing lens doublets that arecolor-corrected at four wavelengths was made without success by R. R.Willey, Jr., as reported in Appl. Opt. 1, 368 (1962). However, Willeydid report that certain pairs of glasses can be used for designing lensdoublets color-corrected at three wavelengths.

An article by M. Gaj and J. Nowak in Optik 29, 321 (1969) stated thatlens triplets color-corrected at three wavelengths could be designedusing various pairs of glasses listed in the optical glass catalogpublished by Glaswerke Schott of Mainz, West Germany. However, two ofthe pairs of glasses identified by Gaj and Nowak were said to require anadditional fluorite lens element to produce the desired color correctionat three wavelengths. Gaj and Nowak did not report any design datanecessary for verifying the design and for constructing lens elementsmade from the optical materials identified in their article.

U.S. Pat. No. 3,674,330 disclosed particular pairs of alkali halidecrystals from which lens doublets that are substantially achromatic overa broad spectrum of wavelengths in the far infrared region, i.e., atwavelengths longer than 1.5 μm can be designed. However, there have beenno reports in the literature of any crystal pairs that can be used fordesigning lens systems color-corrected at more than two wavelengths inthe optical region.

The 1977 article by A. B. Agurok, which is reference (p) on the abovelist, reported on an investigation of the glasses listed in the Russianoptical glass catalog GOST (All Union State Standard) to identify pairsof optical glasses suitable for designing lens doublets color-correctedat more than three wavelengths. The search indicated that no two glassesin the GOST catalog can be combined to make a lens doublet that iscolor-corrected at four wavelengths. Agurok also reported that no twoglasses in the Schott catalog had been found, which in combination makea lens doublet that is color-corrected at four wavelengths.

The various algorithms that were developed in the prior art forselecting optical glasses for designing color-corrected lens systemsmade use of conventional glass parameters (i.e., Abbe numbers andrelative partial dispersion ratios) to characterize the dispersiveproperties of the available types of optical glasses. The approachestaken in the prior art, however, never led to an algorithm of generalvalidity for identifying compatible combinations of optical materialsfor designing lens multiplets that bring more than two wavelengths to acommon focus.

Until the present invention, the problem of correcting optical systemsfor chromatic aberration has been one of the most difficult problemsfacing optical designers. A theoretically rigorous procedure forselecting compatible pairs of optical glasses for designing lensdoublets that are color-corrected at more than two wavelengths haseluded previous researchers. Non-rigorous "cut-and-try" methods havefailed to identify compatible pairs of optical glasses for designinglens doublets color-corrected at more than three wavelengths. C. G.Wynne, in an article entitled "Secondary Spectrum Correction with NormalGlasses", Optics Communications 21, 419 (1977), noted: "It has beengenerally accepted that the correction of secondary spectrumaberrations, in optical imaging systems, necessarily requires the use ofglasses having abnormal relative partial dispersions. This is . . . anerror, arising from defects in the accepted theory of first orderchromatic aberrations."

The inability of the prior art to develop a unified theory for selectingoptical materials to use in designing optical systems that arecolor-corrected at more than two wavelengths can be attributed to theunavailability, unitl now, of a tractable mathematical model forrepresenting optical materials in terms of their dispersioncharacteristics.

DISCLOSURE OF THE INVENTION Buchdahl's Dispersion Equation

The index of refraction (also called the refractive index) of anisotropic homogeneous optical material such as a glass, a crystal or aplastic is a function of the wavelength λ of the optical energy passingthrough the material. Thus, when an optical material is cut and polishedto make a lens element, each wavelength (i.e., each color) is brought toa focus at a different focal position.

Manufacturers of optical glasses conventionally characterize each typeof glass in terms of the refractive index of the glass at certainspecified wavelengths, viz., the wavelengths of the solar emissionlines. In the catalogs published by the leading glass manufacturers,measured values of the refractive index of each glass are listed for thespecified wavelengths. In optical design work, however, it is oftennecessary to know the value of the refractive index of a glass at somewavelength other than those wavelengths specified in the manufacturer'scatalog. The refractive index of a glass at any particular wavelength isdetermined by interpolation from the values given in the catalog byusing a so-called dispersion formula.

A great variety of dispersion formulae have been developed over theyears to express refractive index as a function of wavelength. Acollection of well-known dispersion formulae is given by W. J. Smith inModern Optical Engineering, McGraw-Hill Book Company, Inc., New York(1966), pp. 147-148. In recent years, a power series dispersion formuladerived from a quantum model has come into practically universal useamong optical glass manufacturers throughout the world. This powerseries dispersion formula, which expresses refractive index N in termsof wavelength λ measured in micrometers, is:

    N.sup.2 =A.sub.0 +A.sub.1 λ.sup.2 +A.sub.2 λ.sup.-2 +A.sub.3 λ.sup.-4 +A.sub.4 λ.sup.-6 +A.sub.5 λ.sup.-8, (1)

where the so-called dispersion coefficients A_(j) (j=0, 1, 2, 3, 4, 5),which are provided for each glass by the manufacturer, are obtained by amethod of numerical analysis based upon a least squares fit for measuredvalues of the refractive index averaged over several glass melts. Allmajor optical glass manufacturers provide these dispersion coefficientsA_(j) for each glass, so that optical designers can compute therefractive index of any optical glass for any desired wavelength.

A formidable obstacle to the development of a tractable mathematicalmodel for representing any optical material in terms of its dispersioncharacteristics has been the lack of a dispersion formula in the formN=Σa_(n) x^(n) for predicting the refractive index N, where thecoefficients a_(n) are characteristic of the particular optical materialand are independent of the wavelength λ, and where the variable x is afunction of the wavelength, i.e., x=x(λ). None of the dispersionformulae collected by Smith, op. cit., expresses refractive index inthis mathematically suitable Σa_(n) x^(n) form; and neither does thequantum model dispersion formula of equation (1).

An expression for refractive index, which provided the starting pointfor a mathematical model that made the present invention possible, wasfirst introduced by H. A. Buchdahl in a monograph published by OxfordUniversity Press in 1954. This monograph was subsequently reprinted asOptical Abberration Coefficients, Dover Publications, Inc., New York(1968). Buchdahl's analysis began with an expression for refractiveindex in the form N=N*+c/(λ-λ*), which is known as the Hartmanndispersion formula, where N*, c and λ* are constants that arecharacteristic of the particular optical material. Buchdahl introduced achange of variable from wavelength λ to a so-called color coordinate, orchromatic coordinate, ω, which allowed the refractive index to beexpressed in the form

    N=N.sub.0 +ν.sub.1 ω+ν.sub.2 ω.sup.2 +ν.sub.3 ω.sup.3 + . . . +ν.sub.n ω.sup.n,          (2)

where N₀ is the value of the refractive index at a selected basewavelength λ₀, and where the color coordinate ω is defined as

    ω=δλ/(1+αδλ).        (3)

The quantity δλ is defined as the change in wavelength relative to thebase wavelength, i.e.,

    δλ=λ-λ.sub.0,                   (4)

and the term α is a "universal" constant that is substantiallyindependent of the dispersion characteristics of the particular opticalmaterial. The coefficients, ν_(n), called the dispersion coefficients,are constants characteristic of the particular optical material, and ingeneral vary from one type of optical material to another. P. J. Sandsin J. Opt. Soc. Am. 61, 777 (1971), in which he developed the theory ofhomogeneous lenses, discussed the accuracy of Buchdahl's dispersionformula as given in equation (2).

The color coordinate ω introduced in equation (2) is a function ofwavelength λ, and is "universal" in the sense that it does not involvecharacteristics of the particular optical material. Upon computingvalues of a α for six common optical glasses, Buchdahl found that thevalues of α lie within rather narrow bounds for wavelengths in thevisible spectrum when the base wavelength λ₀ is chosen to be that of thesodium-D line, i.e., 0.5893 μm. This suggested to Buchdahl that α couldbe regarded as a "universal" constant for all glasses for wavelengths inthe visible spectrum. The value of α calculated by Buchdahl was α=2.5.

Ordinarily, the optical materials used for designing lens systems areoptical glasses. For this reason, the analysis presented hereinafterusually refers to optical glasses. However, the analysis is valid foroptical materials generally, and is, therefore, applicable regardless ofwhether the particular material under consideration is a glass or someother material such as, e.g., a crystal (e.g., fluorite or quartz), or aviscous liquid (e.g., immersion oil).

The change in refractive index δN as a function of change in wavelengthδλ relative to the selected base wavelength λ₀, i.e., δN(λ)=N(λ)-N₀, isdefined as the dispersion of the optical material at the wavelength λrelative to the base wavelength λ₀. From equation (2), the dispersion δNof an optical material can be expressed by an equation in the form##EQU1## which will be referred to hereinafter as Buchdahl's dispersionequation.

A detailed analysis of Buchdahl's dispersion equation (5) for all thevarious types of optical glasses listed in the latest catalogs publishedby five leading optical glass manufacturers was undertaken by P. N. Robband R. I. Mercado. The optical glass manufacturers whose catalogs wereused were Schott Optical Glass, Inc., Duryea, Pa.; Ohara Optical GlassMfg. Co., Ltd., Sagamihara, Japan; Hoya Corporation, Tokyo, Japan;Corning-France, Avon, France; and Chance Pilkington Ltd., St. Asaph,Clwyd, U.K. a total of 813 different types of optical glasses wereincluded in the analysis. An announcement of this analysis was publishedin an abstract in J. Opt. Soc. Am. 71 (12), 1638 (1981).

The power series in equation (5) converges very rapidly. The convergencewas found to be so rapid that all 813 glasses studied could bemathematically modelled using only the first three terms of the powerseries, with a resulting maximum absolute error of only 0.0001 in thevisible spectral range from 0.4 μm to 0.7 μm. It is noted that precisionglass melts, on the other hand, can be made to a tolerance of only±0.0002. Thus, a quadratic dispersion model based upon Buchdahl'sdispersion equation, i.e., a dispersion model wherein the degree of thepolynomial in equation (5) is given by n=2, can predict the refractiveindex of an optical glass twice as accurately as the glass can be made.

It was also found that a cubic dispersion model based upon Buchdahl'sdispersion equation, i.e., a dispersion model wherein the degree of thepolynomial in equation (5) is given by n=3, can predict the refractiveindex of an optical glass over an extended spectral range from 0.365 μmto 1.014 μm with an accuracy that is approximately equal to the accuracythat can be maintained in making precision glass melts.

It is remarkable that the value of α, for which Buchdahl adopted the"universal" value of 2.5 on the basis of an analysis of only sixglasses, was found to vary within the narrow range from 2.299 to 2.960,with an average value of 2.577, for a population of 813 glasses.Buchdahl's value of α=2.5 used in defining the color coordinate ω inequation (3) was therefore virtually exact. Buchdahl calculated thedispersion coefficients ν_(i) of equation (5) using refractive indicesfor n wavelengths in an nth-degree polynomial and using a basewavelength of λ₀ =0.5893 μm, which is the wavelength of the yellowsodium-D line. Since equation (5) is linear in the dispersioncoefficients λ_(i), Buchdahl's procedure yields a system of nsimultaneous linear equations in n unknowns. Buchdahl's procedure can bereadily implemented with the aid of a computer.

Values for the dispersion coefficients λ_(i) obtained using Buchdahl'sprocedure were compared with a least-squares solution using sevenwavelenths in the visible range (from 0.4 μm to 0.7 μm) and tenwavelengths in the extended spectral range (from 0.365 μm in theultraviolet to 1.014 μm in the near infrared). It was found that themaximum error could be reduced by a factor of three or more using theleast squares method. Adjustment of the least-squares weights andselection of a base wavelength so as to reduce the maximum error furthergave a total error reduction factor of five.

R. I. Mercado and P. N. Robb announced in an abstract published in J.Opt. Soc. Am. 71 (12), 1639 (1981) that they had developed a method forselecting optical glasses for two-element lens systems whereby paraxialcolor correction could be obtained for three, four and five wavelengths.The glass selection method was indicated to be based on use ofBuchdahl's color coordinate ω rather than on the use of Abbe numbers andrelative partial dispersion ratios. Meanwhile, the method has beengeneralized to enable selection of glasses for designing multipletoptical systems comprising two or more different types of glasses.

Theoretical Basis for the Invention

In the mathematical model leading to the present invention, thedispersive properties of a particular type of optical glass aredescribed in terms of the dispersive power D of the glass. Thedispersive power D is a function of wavelength λ according to therelationship ##EQU2## where N₀ is the value of the refractive index N ofthe glass at the base wavelength λ₀. The base wavelength λ₀ may bearbitrarily selected, but is preferably the wavelength at whichmonochromatic aberrations are calculated for the optical system that isto be designed using the glass.

In developing the theoretical basis for the present invention, thedispersion term δN in equation (6) is replaced by an nth-degreepolynomial in the color coordinate ω in accordance with Buchdahl'sdispersion equation (5). Thus, equation (6) is rewritten as ##EQU3##where the coefficients η_(i), called the η-dispersion coefficients, aregiven by η_(i) =ν_(i) /(N₀ -1). Like the ν-dispersion coefficients ofequation (5), the η-dispersion coefficients of equation (7) areconstants of the particular type of glass and in general vary from onetype of glass to another.

The difference in the dispersive power of a particular type of opticalglass over a wavelength range λ₁ <λ<λ₂ is represented by D(λ₁,λ₂)=D(λ₁)-D(λ₂). Using equation (7), D(λ₁, λ₂) can be expressed in termsof the color coordinate ω by the power series ##EQU4## For the specialcase where λ₁ is hydrogen F line (0.48613 μm), λ₂ is the hydrogen C line(0.65627 μm) and λ₀ is the helium d line (0.58756 μm), it is of interestto note that equation (8) becomes a power series expansion in terms ofthe reciprocal of the Abbe number for the glass.

Thin Lens Equations

Equations (6), (7) and (8) lead to an elegant expression for the totaldioptric power of a lens system comprising a number of thin lenselements in contact. For a single thin lens in air, the dioptric power φis a function of wavelength λ according to the relationship

    φ(λ)=[N(λ)-1](C.sub.1 -C.sub.2),         (9)

where C₁ and C₂ are the two curvatures of the lens. The curvatures C₁and C₂ are independent of the wavelength λ, and are convenientlyrepresented by a constant K=C₁ -C₂ that is characteristic of theconfiguration of the particular lens. Thus φ(λ)=[N(λ)-1]K. For the basewavelength λ₀, the thin lens power is φ(λ₀)=[N₀ -1]K. Since N(λ)=N₀+δN(λ), the thin lens power can be written as

    φ(λ)=φ(λ.sub.0)+KδN(λ). (10)

In terms of the dispersive power D as defined by equation (6), equation(10) can be written in the form

    φ(λ)=φ(λ.sub.0)[1+D(λ)].      (11)

In terms of the color coordinate ω, equation (11) can be expressed as##EQU5## Since ω→0 as λ→λ₀, it follows that φ(λ)→φ(λ₀) as λ→λ₀.

More generally, the total dioptric power Φ of an optical systemcomprising a number of thin lens elements in contact in air is equal tothe sum of the dioptric powers of the individual thin lens elementscomprising the system. Thus, for an optical system comprising k thinlens elements in contact having individual lens powers φ₁, φ₂, . . . ,φ_(k) at the base wavelength λ₀, the total dioptric power Φ of thesystem at the base wavelength λ₀ is ##EQU6## The total dioptric power Φfor any wavelength λ is given by ##EQU7## where φ_(j) is the dioptricpower of the jth lens element in the system at λ₀, and D_(j) (λ) is thedispersive power of the particular type of glass from which the jth lenselement is made. For any wavelength λ, ##EQU8## so that the differencein the total dispersive power of the system over the wavelength range λ₁<λ<λ₂, i.e., D_(j) (λ₁,λ₂)=D_(j) (λ₁)-D_(j) (λ₂), is given by ##EQU9##and the difference in the total dioptric power of the system at thewavelengths λ₁ and λ₂ is ##EQU10##

Thin Lens Doublet

Applying the general equations (13) to (17) for thin lenses in contactto the particular case of a thin lens doublet (i.e., a lens system forwhich k=2), the total dioptric power of the doublet at λ₀ is

    Φ(λ.sub.0)=φ.sub.1 +φ.sub.2.            (18)

From equation (14), the dioptric power of the lens doublet for anywavelength λ is

    Φ(λ)=Φ(λ.sub.0 0+φ.sub.1 D.sub.1 (λ)+φ.sub.2 D.sub.2 (λ).                (19)

In the case of two separated thin lenses in air, the total dioptricpower at λ₀ is

    Φ(λ.sub.0)=φ.sub.1 +(1-k)φ.sub.2.       (20)

where k=dφ₁, where d is the axial separation of the lenses. Theparameter k is a function of wavelength according to the relationshipk(λ)=k[1+D₁ (λ)].

In the case of a lens doublet comprising two different types of glasses,where the two lens elements of the doublet are cemented together, thecondition that the doublet be color-corrected at two wavelengths λ₁ andλ₂ is that Φ(λ₁)=Φ(λ₂). This implies that

    φ.sub.1 D.sub.1 (λ.sub.1,λ.sub.2)+φ.sub.2 D.sub.2 (λ.sub.1,λ.sub.2)=0,                        (21)

where D_(j) (λ₁,λ₂) for j=1, 2 is given by equation (16). Solving thesystem of equations (18) and (21) yields the dioptric powers of theindividual lens elements of the thin lens doublet, to wit:

    φ.sub.1 =-Φ(λ.sub.0)D.sub.2 (λ.sub.1,λ.sub.2)/[D.sub.1 (λ.sub.1,λ.sub.2)-D.sub.2 (λ.sub.1,λ.sub.2)](22)

and

    φ.sub.2 =+Φ(λ.sub.0)D.sub.1 (λ.sub.1,λ.sub.2)/[D.sub.1 (λ.sub.1,λ.sub.2)-D.sub.2 (λ.sub.1,λ.sub.2)]. (23)

The set of equations (22) and (23) is equivalent to the set of equationscommonly used for calculating the dioptric powers of the individual lenselements of a two-glass lens doublet color-corrected at the twowavelengths λ₁ and λ₂, where the dioptric powers of the lens elementscomprising the doublet are expressed in terms of the Abbe numbers forthe glasses from which the lens elements are made.

In the neighborhood of the base wavelength λ₀, the dispersion δN(λ) of atwo-glass lens doublet can be approximated by a linear function in thecolor coordinate ω. Thus, for a two-glass thin lens doublet, equation(5) can be written in the form

    δN(λ)=N(λ.sub.0)+ν.sub.1 ω(λ), (24)

and equations (7) and (8), respectively, become

    D(λ)=η.sub.1 ω(λ)                  (25)

and

    D(λ.sub.1,λ.sub.2)=η.sub.1 [ω(λ.sub.1)-ω(λ.sub.2)].        (26)

Equations (25) and (26) are, respectively, linear models of thedispersive power D(λ) and of the difference in the dispersive powerD(λ₁,λ₂) for the two-glass thin lens doublet over the wavelength rangeλ₁ <λ<λ₂.

Two-Glass Lens Doublet Color-Corrected at Two Wavelengths

In designing a two-glass thin lens doublet that is corrected forparaxial chromatic aberration at two wavelengths, (also called atwo-glass two-color lens doublet), it is assumed that the dispersionproperties of the optical glasses from which the lens elements of thedoublet are made are substantially approximated by equations (25) and(26), i.e., by a linear dispersion model. Thus, each type of opticalglass is uniquely represented by two parameters, viz., the refractiveindex N(λ₀) at the base wave-length λ₀, and the dispersion coefficientη₁ as defined by equation (26). When λ₀ corresponds to the helium dline, and λ₁ and λ₂ are the hydrogen F and C lines, respectivley,equation (26) becomes

    [N(λ.sub.F)-N(λ.sub.C)]/[N(λ.sub.d)-1]=η.sub.1 [ω(λ.sub.F)-ω(λ.sub.C)].        (27)

The left side of equation (27) is the reciprocal of the Abbe number.Since the factor [ω(λ_(F))-ω(λ_(C))] is a constant, equation (27)implies that the dispersion coefficient η₁ of an optical glass isdirectly proportional to the reciprocal of the Abbe number of the glass.The [N(λ₀), η₁ ] glass parameter representation, as suggested by theforegoing analysis, is analogous to the (N_(d), V_(d)) representationcommonly used by optical glass manufacturers for uniquely representingan optical glass, where N_(d) and V_(d) are the refractive index and theAbbe number, respectively, for the glass at the wavelength of the heliumd line.

In terms of the dispersion coefficients η₁ of the glasses from which thelens elements of a two-glass two-color lens doublet are made, thedioptric powers of the individual lens elements at the base wavelengthλ₀ can be expressed by equations in the form

    φ.sub.1 =-η.sub.12 Φ(λ.sub.0)/(η.sub.11 -η.sub.12)                                            (28)

and

    φ.sub.2 =+η.sub.11 Φ(λ.sub.0)/(η.sub.11 -η.sub.12),                                           (29)

where η₁₁ is called the primary dispersion coefficient for the firstglass of the doublet, and η₁₂ is called the primary dispersioncoefficient for the second glass of the doublet. Ordinarily, it isdesirable that the dioptric powers φ₁ and φ₂ of the lens elements of thedoublet be small in value, so that the curvatures of the lens elementswill be small. From equations (28) and (29), it is apparent that theabsolute values of the dioptric powers φ₁ and φ₂ are small when theabsolute value of the factor (η₁₁ -η₁₂) is large. Therefore, to obtain agood dioptric power distribution for a lens doublet, there must be arelatively large difference between the primary dispersion coefficientsof the glasses from which the lens elements of the doublet are made.

The foregoing analysis has been developed with reference to theη-dispersion coefficients. However, inasmuch as ν₁ is directlyproportional to η₁, the analysis is also valid with reference to theν-dispersion coefficients. The technique of the prior art for designinga two-glass lens doublet that is color-corrected at two wavelengths isthus comprehended as a special case of the algorithm developed for thepresent invention, wherein a linear approximation of the power series inBuchdahl's dispersion equation (5) is used.

Two-Glass Lens Doublet Color-Corrected at Three Wavelengths

An extension of the foregoing analysis enables selection of compatibleglasses for designing a two-glass thin lens doublet that is correctedfor axial chromatic aberration at three wavelengths, i.e., a so-calledtwo-glass three-color doublet. To initiate the design of a two-glassthin lens doublet that is color-corrected at wavelengths λ₁, λ₂ and λ₃,where λ₁ <λ₂ <λ₃, the focal length of the doublet may be set equal tounity at the base wavelength λ₀, so that Φ(λ₀)=1 in equation (18). Theconditions that must be satisfied to obtain axial color correction atthe lens doublet at λ₁, λ₂ and λ₃ imply that Φ(λ₁)-Φ(λ₂)=0 andΦ(λ₂)-Φ(λ₃)=0. These conditions lead to a set of homogeneous equationsthat are linear in φ₁ and φ₂. Cast in matrix form, these equations canbe expressed as

    DΦ=0,                                                  (30)

where ##EQU11##

To express equation (30) in terms of the dispersion coefficients of theoptical glasses from which the lens elements comprising the doublet areto be made, a quadratic approximation of the power series in Buchdahl'sdispersion equation (5) is used to obtain axial color correction atthree wavelengths. More generally, the power series in Buchdahl'sdispersion equation (5) is approximated by an (n-1)th degree polynomialin representing a two-glass lens doublet that is color-corrected at nwavelengths.

Assuming that the dispersion properties of each glass of a two-glassthree-color lens doublet (i.e., a two-glass three-color lens systemcomprising k lens elements, where k=1, 2) are approximated withsubstantial accuracy by a quadratic dispersion model, equation (7)becomes

    D.sub.k (λ)=η.sub.1k ω(λ)+η.sub.2k ω.sup.2 (λ),                                 (31)

and equation (8) becomes

    D.sub.k (λ.sub.i,λ.sub.j)=η.sub.1k (ω.sub.i -ω.sub.j)+η.sub.2k (ω.sub.i.sup.2 -ω.sub.j.sup.2). (32)

In designing a two-glass three-color lens doublet, each type of glassused in making the lens elements of the doublet is uniquely representedby the three parameters N₀ (λ), η₁ and η₂, which comprise the [N₀ (λ),η₁, η₂ ] glass parameter representation. Since η₁ and η₂ areproportional to ν₁ and ν₂, respectively, a [N₀ (λ), ν₁, ν₂ ] glassparameter representation in terms of the ν-dispersion coefficients couldalso be used for uniquely representing each type of optical glass.

Using the [N₀ (λ), η₁, η₂ ] glass parameter representation relative tothe base wavelength λ₀, and specifying three wavelengths λ₁, λ₂ and λ₃for which color correction is desired, a system of two nonhomogeneousequations can be obtained from equation (32) for each of the k glassesof the lens doublet, where k=1, 2. This system of equations, however, islinear in the dispersion coefficients η_(1k) and η_(2k). In matrix form,this system of equations may be written as

    Ωη.sub.k =D.sub.k,                               (33)

where ##EQU12## Substituting equation (33) into equation (30) yields

    ΩηΦ=0,                                       (34)

where ##EQU13## The elements of each column vector of the matrix η arethe dispersion coefficients of the glasses from which the lens elementsof the two-glass three-color doublet are made. The matrix element η₁₁ iscalled the primary dispersion coefficient of the first glass of thedoublet, and the matrix element η₂₁ is called the secondary dispersioncoefficient of the first glass of the doublet. Similarly, the matrixelements η₁₂ and η₂₂ are called the primary and secondary dispersioncoefficients, respectively, of the second glass of the doublet.

The matrix Ω in equation (33) may be factored in the following form##EQU14## Since Ω is nonsingular, it has a unique inverse. Hence, ifequation (34) is multiplied by the inverse of Ω, a system of homogeneousequations is obtained in the powers of the constituent lens elements ofthe doublet, where the elements of the coefficient matrix η are thedispersion coefficients of the optical glasses comprising the doublet.Thus,

    ηΦ=0,                                              (36)

which has a nontrivial solution if and only if the determinant of theη-dispersion coefficient matrix vanishes, i.e., if and only if thedeterminant ##EQU15## This condition must be satisfied by the pair ofglasses comprising the doublet, if axial color correction at threewavelengths is to be achieved by the thin lens doublet design. Thus, aparticular pair of optical glasses can be used for designing a two-glassthree-color lens doublet, if and only if the ratios of the primarydispersion coefficient to the secondary dispersion coefficient for bothglasses are substantially equal, i.e., if and only if

    η.sub.11 /η.sub.21 =η.sub.12 /η.sub.12.    (38)

Since n_(ij) =ν_(ij) /[N(λ)-1], equation (37) implies that thedeterminant of the matrix of the corresponding ν-dispersion coefficientsalso vanishes, i.e., that ##EQU16## Thus, using the ν-dispersioncoefficients, the condition that must be satisfied by a pair of opticalglasses in order for those glasses to be compatible with each other in atwo-glass three-color lens doublet can be expressed in terms of theratios of the primary to secondary ν-dispersion coefficients as, viz.,

    ν.sub.11 /ν.sub.21 =ν.sub.12 /ν.sub.22.        (40)

Equation (36) indicates that the dioptric powers of the individual lenselements comprising the two-glass three-color lens doublet are functionsof the dispersion coefficients of the glasses from which the lenselements comprising the doublet are made. The solution of equation (36)can be expressed in terms of the elements of either the first-row vectoror the second-row vector of the η matrix. In terms of the elements ofthe first-row vector,

    φ.sub.1 =+k.sub.1 η.sub.12                         ( 41)

and

    φ.sub.2 =-k.sub.1 η.sub.11,                        (42)

where k₁ is a proportionality factor. For a lens doublet having unitfocal length at the base wavelength λ₀, which is the specified focallength of the two-glass three-color doublet for purposes of thisanalysis,

    k.sub.1 =1/(η.sub.12 -η.sub.11).                   (43)

Similarly, in terms of the elements of the second-row vector,

    φ.sub.1 =+k.sub.2 η.sub.22                         ( 44)

and

    φ.sub.2 =-k.sub.2 η.sub.21,                        (45)

where the proportionality factor k₂ is given by

    k.sub.2 =1/(η.sub.22 -η.sub.21)                    (46)

for the lens doublet having unit focal length at λ₀.

It is instructive to compare equations (41) and (42), which give thedioptric powers of the lens elements elements of a three-color lensdoublet, with equations (28) and (29), which give the dioptric powers ofthe lens elements of a two-color lens doublet. In format, equations (41)and (42) are similar to equations (28) and (29). However, the values ofthe dispersion coefficients η_(ij) obtained from the quadraticdispersion model for the three-color doublet are in general differentfrom the values of the dispersion coefficients η_(ij) obtained from thelinear dispersion model for the two-color doublet.

Two-Glass n-Color Lens Doublet

The theoretical considerations presented above for selecting pairs ofglasses that are compatible for designing two-color and three-color lensdoublets can be generalized and extended mathematically to define theconditions that must be satisfied in order for a pair of glasses to becompatible for designing thin lens doublets color-corrected at nwavelengths, i.e., two-glass n-color lens doublets.

An n-color lens doublet has, by definition, a common paraxial focalpoint at wavelengths λ_(j) for j=1, 2, 3, . . . , n. For an n-color lensdoublet of unit total dioptric power (i.e., Φ=1), at the base wavelengthλ₀, the dioptric powers φ₁ and φ₂ of the individual lens elementscomprising the doublet are related by the expressions

    φ.sub.1 +φ.sub.2 =1                                (47)

and

    DΦ=0,                                                  (48)

where ##EQU17## and O is a column null vector, where λ₁ <λ₂ <λ₃ < . . .<λ_(j-1) <λ_(j) < . . . <λ_(n). Equation (48), which represents a set of(n-1) homogeneous equations that are linear in φ₁ and φ₂, has anontrivial solution if and only if the rank of the coefficient matrix Dis less than 2. The rank of the coefficient matrix D can be determinedby examining all possible determinants of order 2 that can be formed byfixing any two rows of the matrix D. If all possible determinants oforder 2 vanish, the rank of D is less than 2 and a nontrivial solutionof equation (48) exists.

Two-Glass Four-Color Lens Doublet

For a two-glass lens doublet that is color-corrected at four wavelengths(i.e., for n=4), submatrices A, B and C of order 2 obtained from thecoefficient matrix D of equation (48) are given by ##EQU18## A necessarycondition for the existence of a two-glass lens doublet color-correctedat four wavelengths is that the determinants of the submatrices A, B andC vanish identically, which implies that the relationships ##EQU19##must be satisfied by the pair of optical glasses comprising the doublet.

To relate equation (52) to the η-dispersion coefficients of the glasses,a cubic model of the dispersion polynomial is used. Thus, he submatrix Aof equation (49) can be written in the form

    A=Ω.sub.A η                                      (53)

where ##EQU20## The elements of the first and second column vectors ofthe η matrix are the η-dispersion coefficients of the first and secondoptical glasses, respectively, comprising the doublet. The matrixelements η₁₁, η₂₁ and η₃₁ are called the primary, secondary and tertiarydispersion coefficients, respectively, of the first glass of thedoublet; and the matrix elements η₁₂, η₂₂ and η₃₂ are called theprimary, secondary and tertiary dispersion coefficients, respectively,of the second glass of the doublet.

In equation (53), the submatrix A may be considered as the product ofthe factors Ω_(A) and η. From linear algebra, the rank of the submatrixA, written R(A), is related to the ranks of its factors Ω_(A) and η bythe expression

    R(A)≦Minimum [R(Ω.sub.A), R(η)].          (54)

Since the determinant of the submatrix A vanishes as a necessarycondition for the existence of a lens doublet that is color-corrected atfour wavelengths, the rank of A must be less than 2, i.e., R(A)<2. Thecolor coordinates ω_(j) are distinct in value for each wavelength. Sincethe highest-order nonsingular submatrix obtainable from the elements ofthe matrix Ω_(A) is equal to 2, it follows that R(Ω_(A))=2. Since theη-dispersion coefficients are finite in value, R(η)≠0. Furthermore,since R(A)<2, it follows from equation (54) that R(η)<2. Since R(η) musthave an integral value less than 2 and greater than zero, it thereforefollows that R(η)=1.

The fact that the rank of η in equation (53) is equal to unity impliesthat ##EQU21## Thus,

    η.sub.11 /η.sub.21 =η.sub.12 /η.sub.22     ( 56)

    η.sub.11 /η.sub.31 =η.sub.12 /η.sub.32     ( 57)

    η.sub.21 /η.sub.31 =η.sub.22 /η.sub.32,    (58)

which is to say that a particular pair of optical glasses can be used indesigning a two-glass lens doublet that is color-corrected at fourwavelengths, if and only if the ratios of the primary, secondary andtertiary dispersion coefficients of the two glasses are related to eachother in accordance with equations (56), (57) and (58).

Just as the submatrix A in equation (49) was related to the η-dispersioncoefficients by equation (53), so also the submatrix B in equation (50)and the submatrix C in equation (51) can be related to the η-dispersioncoefficients by corresponding equations having the form B=Ω_(B) η andC=Ω_(C) η. These equations for B and C differ from the correspondingequation for A only in the first factor. Thus, an analysis of theequations B=Ω_(B) η and C=Ω_(C) η in the same way as was done above forequation (53) leads to the same result as expressed in equations (56),(57) and (58).

To understand the conditions expressed by equations (56), (57) and (58)in geometrical terms, consider each type of optical glass available tothe designer to be represented by a unique point in a three-dimensionalspace, called a glass space, which is defined by a coordinate systemhaving the η-dispersion coefficients η₁, η₂ and η₃ as orthogonal axes.Two particular glasses G₁ and G₂ having coordinates G₁ (η₁₁, η₂₁, η₃₁)and G₂ (η₁₂, η₂₂, η₃₂), respectively, in the glass space satisfy theconditions of equations (56), (57) and (58), if and only if G₁ and G₂are collinear with the origin of the coordinate system.

The various optical glasses available to the designer can be representedas position vectors in η-space. The conditions of equations (56), (57)and (58), which must be satisfied in designing a two-glass lens doubletthat is color-corrected at four wavelengths, require that the twoglasses G₁ and G₂ selected for making the lens elements of the doubletmust satisfy the vector equation

    G.sub.1 =αG.sub.2,                                   (59)

where G₁ and G₂ are the position vectors for the respective glasses, andα is a real scalar. Since the vectors G₁ and G₂ are collinear, theprojections of G₁ and G₂ on the three orthogonal planes intersecting atthe origin of the coordinate system must also be collinear, which is thegeometric interpretation of the ratios given in equations (56), (57) and(58).

Two-Glass n-Color Lens Multiplet

In a two-glass lens multiplet, there may be any specified number of lenselements, but at least one of the lens elements is made of one type ofglass and at least one other of the lens elements is made of anothertype of glass. A two-glass lens doublet is a special case of a two-glasslens multiplet in which there are only two lens elements.

In designing a two-glass n-color lens doublet, an (n-1)th degree glassdispersion model is used, which is to say that the power series inBuchdahl's dispersion equation (5) is approximated by an (n-1)th degreepolynomial when axial color correction at n wavelengths is desired. Invector representation, each type of optical material available to thedesigner is represented as a unique glass point in an (n-1)-dimensionalvector space having the dispersion coefficients η₁, η₂, η₃, . . . ,η_(n-1) as orthogonal axes. Any two optical materials whose glass pointsare collinear with the origin of the (n-1)-dimensional vector spacecould in principle be used for designing a two-glass lens doublet thatis color-corrected at n wavelengths.

Since axial color correction of a lens multiplet is dependent upon thedispersion properties of the glasses from which the lens elementscomprising the multiplet are made, and does not depend upon the designform (i.e., the number, configuration and arrangement) of the individuallens elements, the technique described above for designing a two-glassn-color lens doublet can be extended for designing a two-glass n-colorlens multiplet comprising more than two lens elements. For purposes ofaxial color correction, a two-glass lens multiplet can be treatedmathematically as a quasi-doublet, where all the individual lenselements made of one type of glass are considered collectively as thefirst lens element of the quasi-doublet, and where all the individuallens elements made of the other type of glass are consideredcollectively as the second lens element of the quasi-doublet. Each ofthe two lens elements of the quasi-doublet is treated as having adioptric power that is the sum of the dioptric powers of all theindividual lens elements made of the same type of glass. Thus, in themathematical analysis of a two-glass lens triplet, quadruplet, orhigher-number multiplet treated as a quasi-doublet, at least one of thelens elements of the quasi-doublet actually comprises two or moreindividual lens elements made of the same type of glass. The totaldioptric power of the quasi-doublet, of course, is the same as the totaldioptric power obtained by summing the dioptric powers of the individuallens elements comprising the multiplet.

Since correction for axial chromatic aberration does not depend upon thedesign form of an optical system, the degree of axial color correctionthat can be obtained for an optical system using a particular pair ofoptical glasses cannot be varied by changing the design form of thesystem. Thus, the same degree of axial color correction can be obtainedfor any lens multiplet using two given optical glasses, regardless ofthe number of lens elements comprising the multiplet, and regardless ofthe configurations and dispositions of the lens elements. However, theremaining optical aberrations, including lateral color, themonochromatic aberrations, and the chromatic variations of themonochromatic aberrations, are sensitive to the design form of a lenssystem.

The selection of compatible optical materials is a necessary conditionfor designing an optical system that is color-corrected at a desirednumber of wavelengths. However, it is not sufficient merely to selectcompatible optical materials. The designer, working with compatibleoptical materials, must further apply his craft with skill in order todetermine a design form that will enable the optical system to meet itsspecified requirements.

A designer can select a compatible pair of optical materials from amongall the various types of optical materials available to him fordesigning a two-glass n-color lens multiplet by:

(a) Determining the dispersion coefficients for each type of opticalmaterial, the dispersion coefficients being coordinates defining aunique "glass point" for each type of optical material in an(n-1)-dimensional "glass space" coordinate system, where n is the numberof wavelengths for which color correction is desired;

(b) Calculating the slope of the hyperline connecting the origin of the"glass space" coordinate system with the "glass point" for each type ofoptical material;

(c) Sorting the various types of optical materials in ascending order ofthe slopes of the hyperlines connecting the "glass points" with theorigin of the coordinate system; and

(d) Selecting a pair of optical materials for which the hyperline slopesare substantially equal.

After identifying the various pairs of optical materials that aresuitable in principle for designing the two-glass n-color lensmultiplet, the designer then selects a preferred pair (or pairs) ofoptical materials to use in implementing the design. Ordinarily, itwould be desirable for the dioptric powers of the individual lenselements comprising the lens multiplet to be minimized, which occurswhen the absolute value of the difference between corresponding primarydispersion coefficients of the two glasses from which the lens elementsare made is a maximum. A further step in the glass selection process,therefore, would ordinarily be to calculate the value of η₁₂ -η₁₁ foreach pair of glasses that could in principle be used for designing thelens multiplet. The preferred pair of glasses is a pair for which thedifference |η₁₂ -η₁₁ | is a maximum.

A particular pair of optical glasses selected according to theabove-described procedure for designing a two-glass three-color lensdoublet might also be found to provide color correction at more thanthree wavelengths. This result can occur, because the conditions forthree-color correction comprise a subset of the conditions for colorcorrection at more than three wavelengths. In general, if two glassvectors G₁ and G₂ are collinear, i.e., if the glass points G₁ and G₂ lieon the same hyperline in (n-1)-dimensional η-space, the projections ofthe vectors G₁ and G₂ on the hyperplane of the next lower degeneracy,i.e., the projections of G₁ and G₂ on the hyperplane in(n-2)-dimensional η-space, must also be collinear with respect to theorigin.

Introduction of Lens Thickness

The foregoing mathematical analysis was developed using the thin lensequations (13) to (17), which represent an idealization that ignores theeffect of lens thickness. In the real world, after a compatible pair ofoptical materials has been selected for designing a two-glasscolor-corrected lens multiplet, the individual lens elements to be madefrom those optical materials must be given suitable thicknesses in orderto be constructible.

In accordance with the present invention, an algorithm has beendeveloped for introducing lens thickness into the design of a two-glasslens multiplet whose "thin lens" elements are to be made from opticalglasses selected according to the above-described procedure. Thealgorithm for lens thickening requires that the powers of the "thinlens" elements be unchanged during the introduction of lens thickness.Also, in order for the "thickened" multiplet to have a total dioptricpower (or focal length) that is identically equal to the total dioptricpower (or focal length) of the idealized thin lens multiplet that itreplaces, the second principal plane of any given lens element of themultiplet must coincide with the first principal plane of an adjacentlens element of the multiplet.

The algorithm for two-glass lens multiplet thickening is described indetail below for the special cases of a cemented lens doublet and anair-spaced lens doublet.

Two-Glass Cemented Lens Doublet

The thick lens dioptric powers φ₁ and φ₂, respectively, of the lenselements comprising a thickened lens doublet are given by the equations##EQU22## where P_(j) (j=1, 2, 3, 4) are the refractive powers of thelens element surfaces in the doublet system, where t₁ and t₂ are theaxial thicknesses of the respective lens elements of the system, andwhere N₁ and N₂ are the refractive indices of the respective glasses atthe base wavelength λ₀.

The refractive power P_(j) of a surface of a lens element is a functionof the surface curvature C_(j) (j=1, 2, 3, 4) and of the refractiveindex of the particular glass from which the lens element is made,according to the equations

    P.sub.1 =C.sub.1 (N.sub.1 -1)                              (62)

    P.sub.2 =C.sub.2 (1-N.sub.1)                               (63)

    P.sub.3 =C.sub.3 (N.sub.2 -1)                              (64)

    P.sub.4 =C.sub.3 (1-N.sub.2).                              (65)

The requirement that adjacent principal planes of the two lens elementsof the doublet coincide implies that ##EQU23## which when solved for P₄yields ##EQU24##

If the two lens elements are cemented together, C₂ =C₃. Thus, equations(63) and (64) can be combined and solved for P₃ to yield ##EQU25##Substituting the expressions for P₄ and P₃ from equations (67) and (68)into equation (61) yields

    φ.sub.2 =a.sub.1 P.sub.1 +a.sub.2 P.sub.2 +a.sub.3 P.sub.1 P.sub.2, (69)

where

    a.sub.1 =-(N.sub.2 t.sub.1 φ.sub.2)/N.sub.1 t.sub.2 φ.sub.1 ; a.sub.2 =(N.sub.2 -1)/(1-N.sub.1);

and

    a.sub.3 =(N.sub.2 -1)t.sub.1 φ.sub.2 /(1-N.sub.1)N.sub.1 φ.sub.1.

Equation (69) and the equations for a₁, a₂ and a₃ can be solvedsimultaneously to yield an equation

    AP.sub.2.sup.2 +BP.sub.2 +C=0,                             (70)

which is quadratic in P₂, where A=a₂ b-a₃ ; B=a₂ -a₁ +a₃ φ₁ -bφ₂ ; C=a₁φ₁ -φ₂ ; and b=-t₁ /N₁. When the discriminant of the quadratic equation(70) is positive, there are two distinct values for P₂. Usually, onlyone of the two values for P₂ can result in a physically realizable lenssystem.

After the values of P₄, P₃ and P₂ have been determined, the values of P₁can be found by simple algebraic substitution in equation (66). Thevalues of the constructional parameters of the lens system arecalculated from equations (62) to (65), after the values of therefractive powers P_(j) have been determined.

Two-Glass Air-Spaced Lens Doublet

The total dioptric power Φ of an air-spaced lens doublet is

    Φ=φ.sub.1 +(1-k)φ.sub.2,                       (71)

where k=dφ₁, where d is the axial separation of the lens elements inair. The procedure used to calculate the dioptric powers φ₁ and φ₂ ofthe individual lens elements of the doublet is identical to theprocedure used for the idealized case of two thin lenses in contact,except that total dioptric power Φ is given by equation (71) and theparameter k varies with wavelength according to the relationship

    k(λ)=k[1+D.sub.1 (λ)],                       (72)

where k is the value of k(λ) at the base wavelength λ₀.

Introduction of thickness into the design of an air-spaced lens doubletfollows the general procedure described above for designing a cementedlens doublet. The individual lens elements are "thickened" by keepingthe powers φ₁ and φ₂ the same, and by keeping the axial separation ofthe adjacent principal planes equal to d.

Two-Glass Mangin Mirror

The glass selection procedure described above is applicable tocatadioptric systems (i.e., systems involving both reflection andrefraction of light), as well as to dioptric systems involving onlyrefraction of light. The simplest form of catadioptric system comprisesa meniscus lens with spherical surfaces, one surface of which isprovided with a reflective coating to form a mirror. Such a "lensmirror" is commonly called a Mangin mirror, after the French militaryofficer who first proposed such imaging systems.

The classical Mangin mirror, which comprises only a negative singletlens made of one optical material, is not corrected for primarychromatic aberration. In order for a Mangin mirror to be useful inimaging applications over a reasonable spectral range, it is necessaryto provide color correction. The requirement for a color-correctedMangin mirror led to the development of the so-called Mangin doubletmirror, which is a compound Mangin mirror in which the classicalnegative meniscus lens is replaced by a doublet lens system in which thetwo lens elements are made of different types of glasses.

Mangin doublet mirrors corrected for axial chromatic aberration at twowavelengths, like ordinary achromatic lens doublets (i.e., two-glasslens doublets corrected at two wavelengths), were known in the priorart. Designs for Mangin doublet mirrors color-corrected at twowavelengths were described by N.v.d.W. Lessing in Appl. Opt. 8, 979(1969), and by S. Rosin and M. Amon in Appl. Opt. 6, 963 (1967) and inU.S. Pat. No. 3,494,688. A method for selecting pairs of optical glassesfor designing cemented Mangin doublet mirrors that are color-correctedat two wavelengths was described by N.v.d.W. Lessing in Appl. Opt. 8,1245 (1969). However, the prior art had no unified theory for selectingoptical materials for designing compound Mangin mirrors color-correctedat more than two wavelengths.

Algorithms developed in the prior art for selecting optical materials tobe used for designing two-glass Mangin mirrors failed to identify anycombinations of glasses or other optical materials that could be usedfor designing any kind of catadioptric system capable of bringing morethan two wavelengths to a common focus. The unified theory leading tothe present invention, however, provides a method for selecting opticalmaterials for designing Mangin mirrors that are color-corrected at morethan two wavelengths.

Two-Glass Mangin Doublet Mirror

A thin cemented Mangin doublet mirror in air, having total catadioptricpower Φ at base wavelength λ₀, can be considered as a system comprisingtwo thin lenses and a mirror in contact. For a marginal paraxial raytrace through the system, the total catadioptric power Φ is thereforegiven by

    Φ=2φ.sub.1 +2φ.sub.2 +φ.sub.m,             (73)

where φ₁ and φ₂ are the respective dioptric powers of the thin lenselements and φ_(m) is the catoptric power of the mirror elementcomprising the system.

At a wavelength other than λ₀, the total catadioptric power Φ of aMangin doublet mirror in terms of the dispersive powers D₁ (λ) and D₂(λ) of the respective thin lens elements, as defined by equation (7), isgiven by

    Φ(λ)=2φ.sub.1 [1+D.sub.1 (λ)]+2φ.sub.2 [1+D.sub.2 (λ)]+φ.sub.m,                                  (74)

which after algebraic simplification becomes

    Φ(λ)-=Φ+2φ.sub.1 D.sub.1 (λ)+2φ.sub.2 D.sub.2 (λ).                                               (75)

The difference in the total catadioptric power of a Mangin doubletmirror at the wavelengths λ_(i) and λ_(j), which is represented asΔΦ(λ_(i),λ_(j))=Φ(λ_(i))-Φ(.lambda._(j)), is given by

    ΔΦ(λ.sub.i,λ.sub.j)=2[φ.sub.1 D.sub.1 (λ.sub.i,λ.sub.j)+φ.sub.2 D.sub.2 (λ.sub.i,λ.sub.j)],                         (76)

where D₁ (λ_(i),λ_(j)) and D₂ (λ_(i),λ_(j)) represent the differencesover the wavelength range from λ_(i) to λ_(j) between the dispersivepowers of the respective glasses from which the lens elements comprisingthe Mangin doublet mirror are made.

Two-Glass Mangin Doublet Mirror Color-Corrected at Three Wavelengths

The conditions that must be satisfied for a two-glass Mangin doubletmirror to be color-corrected at three wavelengths λ₁, λ₂ and λ₃, whereλ₁ <λ₂ <λ₃, imply that ΔΦ(λ₁,λ₂) and ΔΦ(λ₂,λ₃) must both be identicallyequal to zero. In matrix form, this can be written as

    DΦ=0,                                                  (77)

where ##EQU26## Equation (77) is identical to equation (30), whichexpresses the conditions that must be satisfied for a two-glass doubletlens system to be color-corrected at three wavelengths. Thus, assuggested above in the development of equations (31) to (38), in orderfor two particular optical materials to be suitable for designing aMangin doublet mirror that is color-corrected at three wavelengths, itis necessary that the ratio of the primary dispersion coefficient to thesecondary dispersion coefficient be substantially the same for bothoptical materials. Geometrically, the glass points for both opticalmaterials in dispersion coefficient space must lie on a common linethrough the origin of the two-dimensional coordinate system defining thedispersion coefficient space.

The dioptric powers of the two thin lenses comprising the Mangin doubletmirror at the base wavelength λ₀ are given by

    φ.sub.1 =Kη.sub.12                                 ( 78)

and

    φ.sub.2 =-Kη.sub.11,                               (79)

where K is a proportionality factor. Equations (78) and (79) arecorrespondingly similar to equations (41) and (42), respectively;although the proportionality factor K of equations (78) and (79) is ingeneral different from the proportionality factor k₁ of equations (41)and (42). For a Mangin doublet mirror having unit focal length (andtherefore unit total dioptric power) at the base wavelength λ₀, equation(73) becomes

    2φ.sub.1 +2φ.sub.2 +φ.sub.m =1.                (80)

Substituting equations (78) and (79) into equation (80) yields ##EQU27##which enables equations (78) and (79) to be written in the form##EQU28##

Equations (82) and (83) give the dioptric powers at μ₀ of the lenselements of the Mangin doublet mirror as a function of the η-dispersioncoefficients of the lens materials and the catoptric power of the mirrorelement. The catoptric power φ_(m) of the mirror element is independentof wavelength variations, and can be set to a desired value by theoptical designer. For any given value chosen for φ_(m), the powers φ₁and φ₂ of the lens elements are inversely proportional to (η₁₂ -η₁₁).Thus, in order to minimize the values of φ₁ and φ₂, the glass pair usedfor designing the Mangin doublet mirror should be selected so that |η₁₂-η₁₁ | is as large as possible.

In the special case where the mirror element of the Mangin doubletmirror is planar, i.e., where φ_(m) =0, equations (82) and (83) become##EQU29## indicating that the dioptric power distribution of the lenselements of a Mangin doublet mirror that is color-corrected at threewavelengths has a value of one-half the dioptric power distribution ofthe same lens elements used for a two-glass lens doublet that iscolor-corrected at three wavelengths.

In the special case where the catoptric power of the mirror element ofthe Mangin doublet mirror is given by φ_(m) =-1, equations (82) and (83)reduce to ##EQU30## For this particular configuration of the mirrorelement, the dioptric power distribution of the lens elements of aMangin doublet mirror that is color-corrected at three wavelengths isequal to the dioptric power distribution of the same lens elements usedfor a two-glass lens doublet that is color-corrected at threewavelengths.

Two-Glass Mangin Doublet Mirror Color-Corrected at n Wavelengths

The procedure for selecting two different optical materials fordesigning a Mangin doublet mirror that is color-corrected at nwavelengths is the same as the procedure described above for selectingtwo different optical materials for designing a lens doublet that iscolor-corrected at n wavelengths. Each optical material available to thedesigner is represented as a unique glass point G_(q) (q=1, 2) in an(n-1)-dimensional vector space, where the η-dispersion coefficients ofeach particular optical material are the coordinates of the glass pointrepresenting that particular optical material.

A pair of optical materials is suitable for designing a Mangin doubletmirror that is color-corrected at n wavelengths, if and only if theglass points G₁ and G₂ for the two materials obey the vector equation G₁=αG₂. In comparing a Mangin doublet mirror color-corrected at nwavelengths with a lens doublet color-corrected at n wavelengths, wherethe same two types of glasses are used in making the Mangin doubletmirror and the lens doublet, the only difference appearing in themathematical analysis of the system designs would be in the dioptricpower distributions of the thin lens elements.

Two-Glass Mangin Multiplet Mirror Color-Corrected at n Wavelengths

The procedure described above for designing a two-glass Mangin doubletmirror color-corrected at n wavelengths can be generalized for designinga two-glass Mangin multiplet mirror color-corrected at n wavelengths.The two-glass Mangin multiplet mirror can be treated mathematically as atwo-glass Mangin quasi-doublet mirror, where all the individual lenselements made of one type of glass are considered collectively as thefirst lens element of the Mangin quasi-doublet mirror, and all theindividual lens elements made of the other type of glass are consideredcollectively as the second lens element of the Mangin quasi-doubletmirror. The total dioptric power of the first lens element is the sum ofthe dioptric powers of all the individual lens elements made of one typeof glass, and the total dioptric power of the other lens element is thesum of the dioptric powers of all the individual lens elements made ofthe other type of glass. The total catadioptric power Φ of the Manginmultiplet mirror is obtained by generalizing equation (73) into the form##EQU31## where P_(i) (i=1, 2, . . . , r) represents the dioptric powerof the ith lens element made of one type of glass, and Q_(j) (j=1, 2, .. . , s) represents the dioptric power of the jth lens element made ofthe other type of glass. The splitting of the powers of the lenselements of an optical system is a useful technique in optical design toreduce spherochromatism and the higher-order monochromatic aberrations.

Color-Corrected Anamorphic Optical Systems

Each of the optical systems described above is rotationally symmetricalabout a principal axis, called the optical axis, of the system. However,the glass selection procedure of the present invention is generallyapplicable regardless of whether such symmetry exists, and hence is alsoapplicable to anamorphic optical systems comprising, e.g., prisms.

For a thin prism of refractive index N₀ at the base wavelength λ₀ withapex angle α, the angle of deviation β₀ of a ray at wavelength λ₀ afterpassing through the prism is given by

    β.sub.0 =α[N.sub.0 -1].                         (89)

The angle of deviation β of a ray at wavelength λ after passing throughthe prism is given by

    β(λ)=α[N(λ)-1].                   (90)

From equation (6), N(λ)-N₀ =δN=(N₀ -1) D(λ), where D(λ) is thedispersive power of the glass from which the prism is made. Solving for[N(λ)-1] in terms of D(λ), and substituting α=β₀ /(N₀ -1) from equation(89), yields

    β(λ)=β.sub.0 [1+D(λ)].             (91)

In terms of the difference in the dispersive power D(λ_(i),λ_(j)) of theglass over the wavelength range from λ_(i) to λ_(j), as defined above inconnection with equation (8), the chromatic dispersion of the prism fromwhich the glass is made, i.e., Δβ(λ_(i),λ_(j))=β(λ_(i))-β(λ_(j)), isgiven by

    Δβ(λ.sub.i,λ.sub.j)=β.sub.0 D(λ.sub.i,λ.sub.j).                         (92)

Two-Glass Prism System Color-Corrected at Three Wavelengths

Consider a prism system comprising two thin prisms made of differentoptical materials having refractive indices N₁ and N₂, respectively, atthe base wavelength λ₀, and having apex angles α₁ and α₂, respectively.The total angular deviation B of a ray passing through the prism systemis the sum of the angular deviations produced by the individual prismscomprising the system. Thus, at the base wavelength λ₀, the totalangular deviation is

    B=β.sub.1 +β.sub.2,                              (93)

where β₁ =α₁ (N₁ -1) and β₂ =α₂ (N₂ -1).

The conditions for the prism system to have zero chromatic dispersion atλ₁, λ₂ and λ₃, where λ₁ <λ₂ <λ₃, are that Δβ(λ₁,λ₂) and Δβ(λ₂,λ₃) bothidentically equal zero. From equation (92), these conditions can bewritten in matrix form as ##EQU32## Equation (94) is similar to equation(30) for a two-glass lens doublet color-corrected at three wavelengths,and to equation (77) for a two-glass Mangin doublet mirrorcolor-corrected at three wavelengths. Using a quadratic dispersionmodel, the requirement for a particular pair of optical glasses to besuitable for designing a two-glass prism system, whose chromaticdispersion at wavelengths λ₁, λ₂ and λ₃ is zero, is that the determinantof the η-dispersion coefficients of the glasses vanish, i.e., ##EQU33##This implies that the individual thin prisms comprising the prism systemmust have angular deviations given by

    β.sub.1 =kη.sub.12                                ( 96)

and

    β.sub.2 =-kη.sub.11,                              (97)

where k is another proportionality constant. Substituting equations (96)and (97) into equation (93), yields ##EQU34## for the proportionalityconstant. Thus, equations (96) and (97) can be rewritten as ##EQU35##From equations (99) and (100), the apex angles α₁ =β₁ /(N₁ -1) and α₂=β₂ /(N₁ -1) can be written as ##EQU36## Dividing equation (101) byequation (102) yields the ratio ##EQU37## Since η_(ij) =ν_(ij) /(N_(j)-1), equation (103) reduces to

    α.sub.1 /α.sub.2 =-(ν.sub.12 /ν.sub.11), (104)

which indicates that in order for a two-glass prism system to becolor-corrected at three wavelengths, the ratio of the apex angles ofthe individual prisms comprising the system must be inverselyproportional to the ratio of the ν-dispersion coefficients of theoptical materials from which the prisms are made.

Equations (101) and (102) were derived using equations for thin prisms,and hence yield only approximate values. In designing a real prismsystem, the approximate values for α₁ and α₂ provided by thin lenstheory can be adjusted by exact ray-tracing based on Snell's law.

Two-Glass Prism System Color-Corrected at More Than Three Wavelengths

Three-glass prism systems color-corrected at four wavelengths weredescribed by R. E. Stephens in J. Opt. Soc. Am. 56, 213 (1966). However,the prior art did not have a unified theory for selecting opticalmaterials from which such three-glass four-color prisms could bedesigned. No two-glass prism system color-corrected at more than threewavelengths has been reported in the prior art.

In accordance with the present invention, the procedure for selectingoptical materials for designing a two-glass prism system that has zerochromatic dispersion at more than three wavelengths is the same as theprocedure described above for selecting optical materials for designinga two-glass lens system or a two-glass catadioptric system that iscolor-corrected at more than three wavelengths.

The glass selection procedure described above is of general validity,and can be used for designing two-glass optical systems, e.g., lenssystems, catadioptric systems and prism systems, in which colorcorrection at three or more wavelengths is required. Any color-correctedoptical system designed using the glass selection procedure describedabove would therefore be within the scope of the present invention.

Extention of Theory to Include Optical Systems Comprising More Than TwoGlasses

Considering an optical multiplet comprising k thin lens elements incontact made from q different types of glasses, where k≧q, and where themultiplet is color-corrected at n wavelengths, the power seriesexpansion in Buchdahl's dispersion equation (5) is approximated by an(n-1)th degree polynomial in the color coordinate ω. Thus, in equation(7), the terms higher than the (n-1)th degree term are neglected, sothat the dispersive power (D_(q) (λ) for each glass q of the multipletcan be expressed as ##EQU38## where the dispersion coefficients η_(iq)in general vary from glass to glass. Using this (n-1)th degreepolynomial dispersion model, each glass of the multiplet can berepresented by the parameters N_(q) (λ₀), η_(1q), η_(2q), . . . ,η.sub.(n-1)q, where N_(q) (λ₀) is the refractive index of the qth glassat the base wavelength λ₀.

Similarly, the difference in the dispersive power over a wavelengthrange λ₁ <λ<λ₂ for each glass q, represented by D_(q) (λ₁,λ₂)=D_(q)(λ₁)-D_(q) (λ₂), can be expressed in terms of the color coordinate ω bythe power series ##EQU39##

It is noted that an [N_(q) (λ₀), ν_(1q), ν_(2q), . . . , ν.sub.(n-1)q ]glass parameter representation could be used instead of the [N_(q) (λ₀),η_(1q), η_(2q), . . . , η.sub.(n-1)q ] glass parameter representation.However, since the dioptric power of each lens element in the multipletat λ₀ is directly proportional to [N(λ₀)-1], which is used in definingthe η-dispersion coefficients, the η-dispersion coefficients generallylead to more compact and simpler algebraic formulations than theν-dispersion coefficients, and are thus more desirable for rigorousmathematical analysis.

Algorithms for Determining Dispersion Coefficients

The dispersion coefficients η_(iq) for the various types of glasseslisted in the catalogs of the major optical glass manufacturers can becalculated using either equatiion (105) or equation (106).

Since the dispersive power D_(q) (λ) for a particular optical glass q atany wavelength λ can be calculated from ##EQU40## using data provided inthe manufacturer's catalog for that particular glass, it is possible toform a system of (n-1) equations that are linear in the dispersioncoefficients η_(iq) derived from equations (105) and (107), where the(n-1) equations correspond to (n-1) different wavelengths that are alldifferent from the base wavelength λ₀. For each glass q, this system ofnonhomogeneous linear equations can be written in matrix form as

    Ωη.sub.q =D.sub.q,                               (108)

where ##EQU41##

It is a property of equation (108) that the matrix Ω of the colorcoordinates ω can be expressed as the product of two square matrices oforder (n-1), where one of the factors is a diagonal matrix and the otheris a Vandermonde matrix. In factored form, the matrix Ω can be written

    Ω=Diag(ω.sub.1, ω.sub.2, ω.sub.3, . . . ω.sub.n-1)V(ω.sub.1, ω.sub.2, ω.sub.3, . . . ω.sub.n-1),                                         (109)

where ##EQU42## The determinant of Ω is equal to the product of thedeterminants of its factors. Thus, ##EQU43## where II is the poductsymbol.

For a given selection of the base wavelength λ₀ and (n-1) wavelengthsλ_(j), where λ_(j) ≠λ₀ and j=1, 2, . . . , (n-1), the correspondingcolor coordinates ω_(j), where j=1, 2, . . . , (n-1), are distinct andnot equal to zero. This implies that the values of the determinants ofthe factors of Ω can never be zero. Thus, Ω is a nonsingular matrix withrank equal to its order. This further implies that equation (108) has aunique solution, and that the set of values of the dispersioncoefficients η_(iq) for the glasses from which the lens elementscomprising the multiplet are made is unique relative to the nwavelengths λ_(j).

According to the generalized algorithm, therefore, the (n-1)th degreepolynomial as defined by equation (105) is equivalent to a polynomialfit of the dispersive power D_(q) (λ) of the optical glasses for (n-1)wavelengths λ_(j) relative to the base wavelength λ₀. The (n-1)wavelengths λ_(j) can be selected to be the particular wavelengths forwhich color correction is required, i.e., the wavelengths for which themultiplet is designed to have a common focus.

An alternative algorithm for obtaining a set of (n-1) dispersioncoefficients η_(iq) characterizing the dispersion properties of anoptical glass can be developed using equation (106) for the dispersivepower differences relative to wavelength pairs. For each glass, n valuesof its refractive index at n different wavelengths λ_(j), where j=1, 2,. . . , n, are selected from the manufacturer's catalog. Assuming thatλ_(j) ≠λ₀, and that λ₁ <λ₂ < . . . <λ_(n-1) <λ_(n), a system of (n-1)nonhomogeneous equations that are linear in the glass dispersioncoefficients is obtained, from which the values of η_(iq) arecalculated. This system of equations can be written in matrix form as

    ΔΩη.sub.q =Δ.sub.q,                  (111)

where ##EQU44## This system of equations represented by the matrixequation (111) is different from the system of equations represented bythe matrix equation (108) obtained using the first algorithm describedabove. In general, the dispersion coefficients obtained from equation(111) have values that are slightly different from the dispersioncoefficients obtained from equation (108) for the same types of glasses.This second algorithm using equation (111) may be viewed as a polynomialfit of the dispersive power differences of the glasses in terms of (n-1)wavelength pairs, rather than as a polynomial fit of the dispersivepowers in terms of (n-1) wavelengths as in the first algorithm usingequation (108).

Dioptric Powers of Lens Elements

Considering an optical multiplet comprising k cemented thin lenselements in air, where the thin lens elements are made from q differenttypes of glasses, where k≧q, the total dioptric power Φ of the multipletat the base wavelength λ₀ is equal to the sum of the dioptric powers ofthe individual thin lens elements at the same base wavelength λ₀. Thus,##EQU45## where φ_(i) represents the subtotal dioptric power of thegroup of lens elements made of the same ith type of glass. For amultiplet of unit total dioptric power at λ₀ and color-corrected at nwavelengths λ_(j) (j=1, 2, . . . , n), the subtotal dioptric powersφ_(i) at λ₀ are related by the expressions

    φ.sub.1 +φ.sub.2 + . . . +φ.sub.i + . . . +φ.sub.q =1 (113)

and

    ΔΦ(λ.sub.1,λ.sub.2)=ΔΦ(λ.sub.2,.lambda..sub.3)= . . . =ΔΦ(λ.sub.n-1,λ.sub.n)=0, (114)

where ##EQU46## In matrix form, equations (113) and (114) can berepresented as

    GΦ=e.sub.1,                                            (116)

where ##EQU47##

A unique solution of equation (116) exists, if the determinant of the Gmatrix is not equal to zero. From equation (111), the G matrix can bewritten in terms of the dispersion coefficients η_(iq) and the colorcoordinates ω_(q) of the glasses. Expressing G as the product of twomatrices, ##EQU48## where S=(1, 1, . . . , 1) is a sum vector (i.e., avector having unity as a value for each component) of order q, and η isan "(n-1) by q" matrix in the dispersion coefficients given by ##EQU49##and where ΔΩ is the matrix defined above by equation (111), and the 0matrices are null matrices of order (n-1).

Two particular cases are of special interest in considering the dioptricpower distribution of the lens elements, or subgroups of lens elements,of a color-corrected multiplet made of different types of glasses, viz.,the case where n=q, and the case where n>q.

For the case where n=q, equation (116) represents a system of qequations that are linear in the unknown powers φ_(i) (i=1, 2, . . . ,q). Since the number of equations is equal to the number of unknown whenn=q, a unique solution of equation (116) exists when the determinant ofG is not equal to zero. The dioptric powers φ_(i) can then readily becalculated by using Cramer's Rule, or any of a number of other methodsfor solving systems of linear equations.

When G is written as the product of two matrices as in equation (117),the determinant of G is equal to the product of the determinants of itsfactors. Thus, ##EQU50## where ##EQU51##

By Cramer's Rule, the dioptric powers φ_(i) can be expressed as

    φ.sub.i =detM.sub.i /detG                              (120)

for i=1, 2, 3, . . . , n, where M_(i) is the matrix obtained from G byreplacing the ith column of G by the unit column vector e₁.

In terms of the dispersion coefficients η_(iq) of the glasses, thematrix M_(i) can be written as the product of two matrices ##EQU52##where η_(i) is the matrix obtained from the dispersion coefficientsmatrix η when the ith column of η is replaced by a null column vector.Since ##EQU53## equation (120) becomes ##EQU54## where the matrix##EQU55## is obtainable from ##EQU56## by replacing the ith column of##EQU57## with the unit column vector e₁. Thus, ##EQU58##

The determinant of ##EQU59## when expanded in terms of the elements ofthe ith column and its cofactors, yields ##EQU60## where η_(i)(0) is asquare matrix of order (n-1), which is obtained from the rectangularmatrix η by deleting the ith column vector of η.

The determinant of ##EQU61## when expanded in terms of the elements ofits first row and its cofactors, yields ##EQU62## Combining equations(125) and (126) with equation (123) yields ##EQU63## which expresses thedioptric powers φ_(i) of the lens elements, or groups of lens elements,in the multiplet as functions of the dispersion coefficients of thevarious types of glasses from which the lens elements are made.

Dioptric Powers of Two-Glass Lens Multiplet

In the simple case of a two-glass lens multiplet color-corrected at twowavelengths, i.e., where q=n=2, the dispersion model is linear and the ηmatrix in equation (117) reduces to η=[η₁₁ η₁₂ ]. Thus, η₁(0) =[η₁₂ ]and η₂(0) =[η₁₁ ], so that the dioptric powers φ_(i) (i=1, 2) for thetwo glasses, as given by equation (127), are ##EQU64## Equation (128)and (129) are thus seen to be identical to equations (28) and (29)derived above for the case of a two-glass doublet whose total power isequal to unity at the base wavelength.

Dioptric Powers of Three-Glass Lens Multiplet

It is convenient to treat a three-glass lens multiplet as aquasi-triplet (i.e., q=3) in which each effective lens element comprisesall the individual lens elements of the multiplet made from oneparticular type of glass. A particular case of interest is a three-glasstriplet color-corrected at three wavelengths (i.e., k=q=n=3), for whichthe dispersion model is quadratic, so that the η matrix in equation(117) assumes the form ##EQU65## From equation (130), the submatricesη₁(0), η₂(0) and η₃(0) reduce to ##EQU66## which when substituted intoequation (127) yield the dioptric powers

    φ.sub.1 =(η.sub.12 η.sub.23 -η.sub.22 η.sub.13)/d (131)

    φ.sub.2 =-(η.sub.11 η.sub.23 -η.sub.21 η.sub.13)/d (132)

    φ.sub.3 =(η.sub.11 η.sub.22 -η.sub.21 η.sub.12)/d, (133)

where ##EQU67## i.e., d=η₁₁ (η₂₂ -η₂₃)+η₁₂ (η₂₃ -η₂₁)+η₁₃ (η₂₁ -η₂₂).

Four-Glass Lens Multiplet

Another particular case of interest is a four-glass lens quadruplet thatis color-corrected at four wavelengths, i.e., k=q=n=4. Such a quadrupletwas called a superachromatic quadruplet by N.v.d.W. Lessing in Appl.Opt. 9, 2390 (1970). Although various combinations of four differenttypes of optical glasses suitable for designing such superachromaticquadruplets were reported in the prior art, a general technique foridentifying all the optical materials that could be used for designingsuch superachromatic quadruplets was not developed until the presentinvention.

In accordance with the present invention, a cubic dispersion model isused to select glasses for designing a superachromatic quadruplet.Accordingly, the η matrix in equation (117) assumes the form ##EQU68##from which the submatrices η_(j)(0) are obtained, and the dioptric powerdistributions φ_(i) (i=1, 2, 3, 4) are calculated from equation (127).

Although lens multiplets comprising q different types of optical glassescolor-corrected at n=q wavelengths were known in the prior art for q=2,3, 4, the method of the present invention whereby suitable opticalmaterials for designing such multiplets can be identified by expressingthe dispersion characteristics of the optical materials in terms ofη-dispersion coefficients is completely novel. Furthermore, it is also anovel feature of present invention that the practicality of using anyparticular combination of optical materials to design a lens multipletcan be determined a prior by expressing the dioptric powers of theindividual lens elements of the multiplet as functions of theη-dispersion coefficients of the optical materials selected for makingthe lens elements.

Special Case for n>q

Equation (116), G Φ=e₁, represents a system of n equations that arelinear in the unknown dioptric powers φ_(i) (i=1, 2, . . . , q). For thecase where n>q, the number of equations would be greater than the numberof unknowns. For k=q=2, i.e., for a two-glass doublet color-corrected atn wavelengths, equation (116) becomes ##EQU69## which is the matrixequation equivalent of equations (47) and (48) for a two-glass n-colorlens doublet.

In general, for a lens multiplet made of q different types of glasses,the matrix D in equation (48) is an "(n-1)×q" rectangular matrix inD_(q) (λ_(j-1),λ_(j)). The conditions that must be satisfied by theindividual glasses are obtained in the same way as for the doublet case,i.e., by examining the rank of the matrix D. Thus, a set of q glassescould be used for designing a lens multiplet that is color-corrected atn wavelengths, if and only if the rank of the matrix D for the set ofglasses is less than q.

Special Case for n=q+1

An interesting special case, which is instructive in describing theinvention, is a lens multiplet comprising q different types of opticalglasses color-corrected at q+1 wavelengths. For this case, equation(116) becomes ##EQU70##

Equation (136) is equivalent to the sets of equations

    φ.sub.1,φ.sub.2 + . . . +φ.sub.q =1            (137)

and

    DΦ=0,                                                  (138)

where ##EQU71## and 0 is a null column vector of order q.

In terms of the chromatic coordinates and the dispersion coefficients ofthe glasses, the homogeneous matrix equation (138) can be written as

    ΔΩηΦ=0,                                (139)

where ΔΩ is defined as in equation (111), and η is the matrix of thedispersion coefficients as defined in equation (117). Both ΔΩ and η aresquare matrices of order q. The matrix ΔΩ is nonsingular, so thatequation (139) reduces to

    ηΦ=0.                                              (140)

In order for equation (140) to have a nontrivial solution, it isnecessary that the determinant of η vanish, i.e., that ##EQU72## Thus,equation (141) represents the condition that must be satisfied by theη-dispersion coefficients of the q different types of glasses used indesigning the lens multiplet in order for the multiplet to becolor-corrected at q+1 wavelengths.

The general solution of the homogeneous equation (140) is

    φ.sub.j =αN.sub.ij                               ( 142)

for j=1, 2, . . . , q, where α is an arbitrary constant and the N_(ij)are the cofactors of the elements of the ith row of the η matrix. Theconstant α can be chosen to normalize the total dioptric power of thelens multiplet at the base wavelength λ₀. Substituting equation (142)into equation (137) yields ##EQU73##

The set of equations (142) and (143) enables the dioptric powers of theq different groups of lens elements corresponding to the q differenttypes of glasses used in making the lens multiplet to be determined,where the lens multiplet is color-corrected at q+1 wavelengths. Theabove analysis for the case where n=q+1 holds true in the particularcase where there is only one lens element in each of the groups of lenselements, i.e., where k=q.

When k=2, where k=q, there are only two lens elements, with each lenselement being made of a different type of glass. Thus, equation (141)reduces to equation (37), and equation (142) reduces to equations (41)and (42), which were derived above for the case of a two-glass lensdoublet color-corrected at three wavelengths.

When k=3, where k=q, the resulting lens multiplet is a three-glasstriplet. In the case under consideration, the lens multipletcolor-corrected at n=q+1 wavelengths becomes a three-glass lens tripletcolor-corrected at four wavelengths, for which the designation"superachromat" was used by Herzberger in U.S. Pat. No. 3,395,962. Thepresent invention provides a technique for identifying all possibleglass combinations that could be used in designing such superachromaticlens triplets, and for selecting practical three-glass combinations interms of dioptric power distributions.

For k=q=3, equation (141) reduces to ##EQU74## and the dioptric powersof the individual lens elements, as given by equation (142), become

    φ.sub.1 =α.sub.1 (η.sub.12 η.sub.23 -η.sub.22 η.sub.13)                                             (145)

    φ.sub.2 =-α.sub.1 (η.sub.11 η.sub.23 -η.sub.21 η.sub.13)                                             (146)

    φ.sub.3 =α.sub.1 (η.sub.11 η.sub.22 -η.sub.21 η.sub.12),                                            (147)

where

    α.sub.1 =1/[η.sub.11 (η.sub.22 -η.sub.23)+η.sub.12 (η.sub.23 -η.sub.21)+η.sub.13 (η.sub.21 -η.sub.22)].

The expressions for the dioptric powers φ₁, φ₂ and φ₃ given in equations(145), (146) and (147) are identical to those in equations (131), (132)and (133) for the case of a three-glass lens triplet color-corrected atthree wavelengths. However, a three-glass lens triplet color-correctedat four wavelengths must satisfy the additional conditions implied inequation (144). The conditions of equation (144) can be interpretedgeometrically by considering each glass available to the designer as aunique point in a three-dimensional glass space defined by the axes η₁,η₂ and η₃. Three glasses, represented by the points G₁, G₂ and G₃,satisfy the conditions of equation (144), if and only if G₁, G₂ and G₃lie in a common plane that passes through the origin of the coordinatesystem.

Applying a series of elementary row and column operations to the matrixof equation (144) results in ##EQU75## Defining μ₁ =η₁ /η₃ and μ₂ =η₂/η₃, so that μ_(1k) =η_(1k) /η_(3k) and μ_(2k) =η_(2k) /η_(3k) for k=1,2, 3, the parameters μ₁ and μ₂ can be considered as axes of atwo-dimensional rectangular coordinate system into which thethree-dimensional coordinate system defined by the axes η₁, η₂ and η₃ istransformed. Equation (148) implies that when the various opticalmaterials available to the designer are plotted as points in thetwo-dimensional coordinate system with axes μ₁ and μ₂, the glass pointsG₁ (μ₁₁,μ₂₁), G₂ (μ₁₂,μ₂₂) and G₃ (μ₁₃,μ₂₃) are collinear. Thistwo-dimensional geometrical representation of the glasses in (μ₁,μ₂)space is more convenient than the three-dimensional representation in(η₁,η₂,η₃) space. With the aid of a digital computer, it is relativelyeasy to obtain the coordinates for the glasses in (μ₁,μ₂) space from theknown values of the n-dispersion coefficients of the optical materialsin (η₁,η₂,η₃) space.

When k=4, where k=q, the resulting lens multiplet is a four-glassquadruplet. The case under consideration of a lens multipletcolor-corrected at n=q+1 wavelengths becomes a four-glass lensquadruplet color-corrected at five wavelengths. The dispersion model isquartic, and equation (141) becomes ##EQU76## Equation (149) expressesthe conditions that must be satisfied by a set of four optical materialsin order for those four optical materials to be compatible with eachother for use in designing a five-color lens quadruplet. The dioptricpowers of the optical materials selected for designing the five-colorlens quadruplet can be calculated from equation (142).

Geometrically, equation (149) implies that if each optical materialavailable to the designer is represented by a unique point in afour-dimensional n-dispersion coefficient space with rectangularcoordinate axes η₁, η₂, η₃ and η₄, the glass points, G₁, G₂, G₃ and G₄must lie in a common hyperplane that passes through the origin of thecoordinate system.

A more easily visualizable geometrical interpretation of equation (149)can be obtained by applying a series of elementary row and columnoperations to the matrix of equation (149), which results in thetransformed equation ##EQU77## where μ_(1k) =η_(1k) /η_(4k), μ_(2k)=η_(2k) /η_(4k) and μ_(3k) =η_(3k) /η_(4k) for k=1, 2, 3, 4.

It is noted that the definition of the parameter μ_(ik) (i=1, 2, 3) forthe case of a four-glass five-color lens quadruplet is different fromthe definition of the parameter μ_(i) (i=1, 2) introduced above for thecase of a three-glass four-color lens triplet.

Equation (150) implies that if each optical material available to thedesigner is represented by a unique point in a three-dimensionalrectangular coordinate system with axes μ₁, μ₂ and μ₃, the four glasseswith coordinates G₁ (μ₁₁,μ₂₁,μ₃₁), G₂ (μ₁₂,μ₂₂,μ₃₂), G₃ (μ₁₃,μ₂₃,μ₃₃)and G₄ (μ₁₄,μ₂₄,μ₃₄) must be collinear in the three-dimensional(μ₁,μ₂,μ₃) space in order to be suitable for designing a four-glassfive-color lens quadruplet. With the aid of a digital computer, it isrelatively easy to obtain the coordinates of the optical materials inthe three dimensional (μ₁,μ₂,μ₃) space from the unknown values of theη-dispersion coefficients of those materials in the four dimensional(η₁,η₂,η₃,η₄) space.

Method for Selecting Three Compatible Glasses

In selecting a combination of three different optical materials fordesigning a "three-glass" multiplet (i.e., an optical system made ofthree different optical materials, whether of glass or otherwise) thatis color-corrected at four wavelengths, it is more convenient to use atwo-dimensional glass point representation in (μ₁,μ₂) space rather thana three-dimensional glass point representation in (μ₁,μ₂,μ₃) space foreach of the available optical materials.

The selection process starts with the choice of a particular opticalmaterial represented by glass point G₁ (μ₁₁,μ₂₁). The origin of thecoordinate system defined by the axes μ₁ and μ₂ is then translated toG₁, so that the "new" axes μ₁ ' and μ₂ ' (i.e., the axes aftertranslation) are parallel to the original axes μ₁ and μ₂, respectively.The coordinates of each optical material are then expressed in terms ofthe translated axes μ₁ ' and μ₂ ', so that G_(k) (μ_(1k),μ_(2k)) isexpressed as G_(k) '(μ_(1k) ',μ_(2k) '), where

    μ.sub.1k' =μ.sub.1k -μ.sub.11

    μ.sub.2k' =μ.sub.2k -μ.sub.21

, (151 )

which are the equations for transformation of coordinates.

The equation of a straight line through the origin G₁ '(0,0) of thetransformed coordinate system defined by the translated axes μ₁ ' and μ₂' is given by μ₂ '=m' μ₁ ', where m' is the slope of the line referredto the translated axes. Thus, a straight line passing through G₁ '(0,0)and a glass point G_(k) '(μ_(1k) ',μ_(2k) ') has a slope

    m'=μ.sub.2k '/μ.sub.1k ',                            (152)

which is the equation used to determine the slope m' for each availableoptical material. The arc tangent of m' is the slope angle of the glasspoint G_(k) ' relative to the μ₁ ' axis.

Using equation (152), a table is then made in which all availableoptical materials are sorted in ascending or descending order of thevalues of m'. Any pair of optical materials having substantially thesame value of m' (or, equivalently, having substantially the same valueof the arc tangent of m') as the orginally chosen optical material,whose glass point G₁ is the origin of the translated axes, can be usedin combination with that originally chosen optical material fordesigning a "three-glass" optical system that is color-corrected at fourwavelengths.

Method for Selecting Four Compatible Glasses

In selecting a combination of four different optical materials fordesigning a "four-glass" multiplet that is color-corrected at fivewavelengths, it is more convenient to use a three-dimensional glasspoint representation in (μ₁,μ₂,μ₃) space rather than a four-dimensionalglass point representation in (η₁,η₂,η₃,η₄) space for each of theavailable optical materials.

The selection process starts by choosing two optical materialsrepresented by glass points G₁ (μ₁₁,μ₂₁,μ₃₁) and G₂ (μ₁₂,μ₂₂,μ₃₂),respectively. The origin of the coordinate system defined by theorthogonal axes μ₁, μ₂ and μ₃ is then translated to G₁, so that thetranslated axes μ₁ ', μ₂ ' and μ₃ ' are parallel to the original axesμ₁, μ₂ and μ₃, respectively. The coordinates of each optical materialare then expressed in terms of the translated axes, so that G_(k)(μ_(1k),μ_(2k),μ_(3k)) is expressed as G_(k) '(μ_(1k) ',μ_(2k) ',μ_(3k)'), where ##EQU78##

The equation of a straight line through the origin G₁ '(0,0,0) of thetransformed coordinate system defined by the translated axes μ₁ ', μ₂ 'and μ₃ ' is given by μ₁ '/μ₁₂ '=μ₂ '/μ₂₂ '=μ₃ '/μ₃₂ ', which isequivalent to the system of equations ##EQU79## It is noted that anyvalues for μ₁, μ₂ and μ₃ satisfying any two of the equations (154) alsosatisfy the third equation. Equations (154) also imply that ##EQU80##where k is a constant different from zero. Therefore, the glass pointsG₂ '(μ₁₂ ',μ₂₂ ',μ₃₂ '), G₃ '(μ₁₃ ',μ₂₃ ',μ₃₃ ') and G₄ '(μ₁₄ ',μ₂₄',μ₃₄ ') are collinear with the origin G₁ '(0,0,0), if and only if theircorresponding coordinates are proportional to each other. Opticalmaterials represented by glass points G₃ ' and G₄ ' can be selected fromamong those optical materials having coordinates that are substantiallyproportional to the corresponding coordinates of G₂ '.

DESCRIPTION OF THE DRAWING

FIG. 1 is a geometrical representation in dispersion coefficient spaceof an (n-1)th degree dispersion model for an optical system comprisingtwo different optical materials.

FIG. 2 is a geometrical representation in dispersion coefficient spaceof a linear dispersion model for an optical system comprising twodifferent optical materials.

FIG. 3 is a geometrical representation in dispersion coefficient spaceof a quadratic dispersion model for an optical system comprising twodifferent optical materials.

FIG. 4 is a geometrical representation in dispersion coefficient spaceof a cubic dispersion model for an optical system comprising twodifferent optical materials.

FIG. 5 is a geometrical representation in dispersion coefficient spaceof a cubic dispersion model for an optical system comprising threedifferent optical materials.

FIG. 6A is a plot in the (η₁,η₂) plane of the η-dispersion coefficientsfor 813 different types of optical glasses listed in the catalogs offive major optical glass manufacturers.

FIG. 6B is an enlargement of a portion of the plot shown in FIG. 6A.

FIG. 7A is a plot in the (ν₁,ν₂) plane of the ν-dispersion coefficientsfor the same 813 different types of optical glasses whose η-dispersioncoefficients are shown in FIG. 6A.

FIG. 7B is an enlargement of a portion of the plot shown in FIG. 7A.

FIG. 8 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a two-color cemented lens doublet havingunit focal length at λ₀ =0.54607 μm for a relative aperture of f/10,where the lens elements comprising the doublet are made of Schott BK7and Schott F2 glasses.

FIG. 9 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a three-color cemented lens doublet of unitfocal length at λ₀ =0.54607 μm for a relative aperture of f/10, wherethe lens elements comprising the doublet are made of Schott FK51 andSchott K51 glasses.

FIG. 10 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color cemented lens doublet of unitfocal length at λ₀ =0.54607 μm for a relative aperture of f/10, wherethe lens elements comprising the doublet are made of Corning-France486817 and Ohara LAK14 glasses.

FIG. 11 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color air-spaced lens doublet ofunit focal length at λ₀ =0.54607 μm for a relative aperture of f/10,where the lens elements comprising the doublet are made of the sametypes of glasses as for the lens doublet of FIG. 10.

FIG. 12 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a five-color air-spaced lens doublet ofunit focal length at λ₀ =0.54607 μm for a relative aperture of f/10,where the lens elements comprising the doublet are made of Schott PK51and Hoya LAC7 glasses.

FIG. 13 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color cemented two-glass lenstriplet having a focal length that is 40 times the focal length of thedoublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative apertureof f/10, where the first and third lens elements are made ofCorning-France 486817 glass and the second lens element is made ofCorning-France 589612 glass.

FIG. 14 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color two-glass lens triplet havinga focal length that is 40 times the focal length of the doublets shownin FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/10, wherethe lens elements are made of the same types of glasses as for the lenstriplet shown in FIG. 13, the first and second lens elements arecemented together, and an air gap is provided between the second andthird lens elements.

FIG. 15 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color cemented two-glass lensquadruplet having a focal length that is 40 times the focal length ofthe doublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relativeaperture of f/10, where the first and third lens elements are made ofCorning-France 486817 glass, and the second and fourth lens elements aremade of Corning-France 589612 glass.

FIG. 16 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color two-glass lens quadruplethaving a focal length that is 40 times the focal length of the doubletsshown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/10,where the lens elements are made of the same types of glasses as for thequadruplet shown in FIG. 15, except that the first and fourth lenselements are made of Corning-France 486817 glass, and the second andthird lens elements are made of Corning-France 589612 glass, where thefirst and second lens elements are cemented together, the third andfourth lens elements are cemented together, and an air gap is providedbetween the second and third lens elements.

FIG. 17 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color two-glass lens quadruplethaving a focal length that is 40 times the focal length of the lensdoublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative apertureof f/10, where the lens elements are made of the same types of glassesas for the quadruplet show in FIG. 15, where the first and third lenselements are made of Corning-France 486817 glass, and the second andfourth lens elements are made of Corning-France 589612 glass, where thefirst and second lens elements are cemented together, the third andfourth lens elements are cemented together, and an air gap is providedbetween the second and third lens elements.

FIG. 18 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color collimator comprising atwo-glass lens quadruplet, where the two glasses from which the lenselements of the quadruplet are made are Hoya FCD10 glass and Ohara LAK14glass.

FIG. 19 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a five-color collimator comprising atwo-glass lens quadruplet, where the two glasses from which the lenselements of the quadruplet are made are Schott PK51 glass and Hoya LAC7glass.

FIG. 20 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a five-color collimator comprising a lensquadruplet made of one type of optical glass and one type of crystal,where the glass is Hoya LAC7 glass and the crystal is calcium fluoride.

FIG. 21 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a five-color two-glass cemented Mangindoublet mirror system of unit focal length at λ₀ =0.54607 μm for arelative aperture of f/2, where the system elements comprise a firstlens element made of Hoya FDC10 glass and a second lens element made ofHoya FC3 glass, the outward facing surface of the second lens elementhaving a reflective coating.

FIG. 22 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color two-glass cemented Mangindoublet mirror system of unit focal length at λ₀ =0.54607 μm for arelative aperture of f/2, where the system elements comprise a firstlens element made of Hoya BAF11 glass and a second lens element made ofHoya FL6 glass, the outward facing surface of the second lens elementhaving a reflective coating.

FIG. 23 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color two-glass cemented Mangindoublet mirror system of unit focal length at λ₀ =0.54607 μm for arelative aperture of f/2, where the system elements comprise a firstlens element made of Corning-France 486817 glass and a second lenselement made of Corning-France 697554 glass, the outward facing surfaceof the second lens element having a reflective coating.

FIG. 24 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a five-color two-glass air-spaced Mangindoublet mirror system of unit focal length at λ₀ =0.54607 μm for arelative aperture of f/2, where the system elements comprise a firstlens element made of Schott PK51 glass and a second lens element made ofHoya LAC7 glass, the outward facing surface of the second lens elementhaving a reflective coating.

FIG. 25 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a five-color two-glass air spaced Mangindoublet mirror system of unit focal length at λ₀ =0.54607 μm for arelative aperture of f/2, where the system elements comprise a firstlens element made of Hoya FCD10 glass and a second lens element made ofOhara PSK1 glass, the outward facing surface of the second lens elementhaving a reflective coating.

FIG. 26 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-color two-glass air spaced Mangindoublet mirror system of unit focal length at λ₀ =0.54607 μm for arelative aperture of f/2, where the system elements comprise a firstlens element made of Hoya FCD10 glass and a second lens element made ofHoya LAC14 glass, the outward facing surface of the second lens elementhaving a reflective coating.

FIG. 27 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a two-glass five-color Petzval lens systemhaving a focal length of 600 mm at λ₀ =0.54607 μm for a relativeaperture of f/3.5 and an angular field of ±6°.

FIG. 28 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a two-glass four-color Petzval lens systemhaving a focal length of 360 mm at λ₀ =0.54607 μm for a relativeaperture of f/3.5 and an angular field of ±6°.

FIG. 29 provides a profile drawing a plot of paraxial marginal rayheight versus wavelength for a two-glass three-color Petzval lens systemhaving a focal length of 1000 mm at λ₀ =0.54607 μm for a relativeaperture of f/2 and an angular field of ±2.1°.

FIG. 30 provides a profile drawing and a plot of change in totaldeviation angle versus wavelength for a two-glass five-color doubletprism system having a total deviation angle of 6°.

FIG. 31 provides a profile drawing and a plot of change in totaldeviation angle versus wavelength for a two-glass five-color quintupletprism system having a total deviation angle of 30°, where the apexangles of the first, second, third and fourth prism elements are equalto each other in absolute value, and where the apexes of the second andfourth prism elements have an orientation that is opposite to theorientation of the apexes of the first and third prism elements.

FIG. 32 provides a profile drawing and a plot of change in totaldeviation angle versus wavelength for a two-glass five-color quintupletprism system having a total deviation angle of 30°, where the prismelements are made of the same types of glasses as for the correspondingprism elements of the prism system shown in FIG. 31, but where the apexangles of the five prism elements are all different in value from eachother.

FIG. 33 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a three-glass lens quintuplet that iscolor-corrected at five wavelengths, the quintuplet having a focallength of 1000 mm at λ₀ =0.54607 μm for a relative aperture of f/10, andcomprising a first lens element made of Ohara FKO1 glass, a second lenselement made of Ohara KZFS8 glass and air-spaced from the first lenselement, a third lens element made of Ohara FKO1 glass and cemented tothe second lens element, a fourth lens element made of Ohara SF15 glassand air-spaced from the third lens element, and a fifth lens elementmade of Ohara KZFS8 glass and cemented to the fourth lens element.

FIG. 34 provides a profile drawing and a plot of paraxial marginal rayheight versus wavelength for a four-glass lens quintuplet that iscolor-corrected at five wavelengths, the quintuplet having a focallength of 1000 mm at λ₀ =0.54607 μm for a relative aperture of f/10, andcomprising a first lens element made of Schott PK51 glass, a second lenselement made of Schott LAK20 glass and air-spaced from the first lenselement, and third lens element made of Schott PK51 glass and cementedto the second lens element, a fourth lens element made of Schott KZFSN4glass and air-spaced from the third lens element, and a fifth lenselement made of Schott KSFS6 glass and cemented to the fourth lenselement.

BEST MODE OF CARRYING OUT THE INVENTION

It is a feature of the present invention that combinations of opticalglasses, which are compatible with each other for designing opticalsystems that are color-corrected at three or more wavelengths, can bereadily selected from among the various optical glasses available to thedesigner.

Commercially available types of optical glasses are listed in catalogspublished by various manufacturers. The latest edition (1980 Edition) ofthe Schott catalog lists 238 different types of optical glasses. Thelatest Ohara catalog (distributed in 1981) lists 227 different types ofoptical glasses. The latest Hoya catalog (distributed in 1981) lists 206different optical glass types. The latest Corning-France catalog (1980Edition) lists 80 different optical glass types. The latest ChancePilkington catalog (1980 Edition) lists 62 different optical glasstypes. These five manufacturers presently produce a total of 813different types of optical glasses.

The initial problem in designing an optical system that is to becolor-corrected at more than two wavelengths is to select a suitablecombination of optical materials from among the various types of opticalmaterials available to the designer. The present invention provides aunified procedure for selecting suitable combinations of opticalmaterials for use in designing color-corrected optical systems. Theprocedure of the present invention can be conveniently illustrated inconnection with the design of a two-glass lens doublet (i.e., amultiplet comprising two lens elements made from different types ofglasses), which is color-corrected at n different wavelengths, wheren>2. The principles discussed herein apply as well to the selection ofglasses or other optical materials for designing color-corrected lenstriplets and higher-number lens multiplets.

The number of possible pairs of glasses that can be formed from the 813different types of optical glasses listed in the catalogs published bythe leading optical glass manufacturers can be determined by calculatingthe combination of n things taken r at a time, i.e., C(n,r)=n!/r!(n-r)!,where n=813 and r=2. This number, which is independent of the order inwhich the glasses are selected, is 330,078. Thus, if a "cut-and-try"approach were used to identify pairs of optical glasses that arecompatible with each other for designing a two-glass color-correctedlens doublet, more than 330,000 different combinations of glasses wouldhave to be investigated.

Very few pairs of optical glasses suitable for designing a two-glasslens doublet that is color-corrected at three wavelengths have beenreported in the literature; and those few suitable glass pairs that havebeen reported were discovered by cut-and-try techniques without thebenefit of a unified theoretical basis for selecting the particularglasses. Until the present invention, there has been no systematic wayof knowing whether any particular glass could be paired with any otherparticular glass to produce an optical system that is color-corrected atmore than two wavelengths.

In FIG. 1, two optical materials G₁ and G₂ are represented bycorresponding glass points G₁ [η₁₁, η₂₁, η₃₁, . . . , η.sub.(n-1)1 ] andG₂ [η₁₂, η₂₂, η₃₂, . . . , η.sub.(n-1)2 ] in an (n-1)-dimensionalorthogonal coordinate system defining a so-called glass space. Thecoordinates of any particular optical material in the glass space arethe η-dispersion coefficients of the optical material, as derived from ageneralized (n-1)th degree dispersion model. The condition that must besatisfied in order for a two-glass optical system comprising these twoparticular optical materials G₁ and G₂ to be color-corrected at nwavelengths is that the vectors G₁ and G₂ in the glass space satisfy thevector equation G₁ =αG₂, whe α is a real scalar quantity. In graphicterms, as illustrated in FIG. 1, the glass points G.sub. 1 and G₂ mustbe substantially collinear with the origin of the coordinate system.

For the special case of a two-glass optical system that iscolor-corrected at two wavelengths (i.e., where n=2), the dispersionmodel used to derive the η-dispersion coefficients is linear. As shownin FIG. 2, a two-glass two-color optical system can be designed usingany pair of optical glasses, because the glass points for all opticalglasses necessarily lie on the same straight line in the one-dimensionalη-dispersion coefficient space of the linear dispersion model. Although,any two different optical glasses could in principle be used incombination with each other for designing an optical system that iscolor-corrected at two wavelengths, it would normally be desirable touse only those pairs of glasses for which the absolute value of thedifference between their corresponding primary dispersion coefficientsis as large as possible.

For the special case of a two-glass optical system that iscolor-corrected at three wavelengths (i.e., where n=3), the dispersionmodel used to derive the η-dispersion coefficients is quadratic. Asshown in FIG. 3, a two-glass three-color optical system can be designedusing a pair of optical glasses whose glass points are collinear withthe origin of the coordinate system defining the two-dimensionalη-dispersion coefficient space of the quadratic dispersion model. Thus,if the glass points for the various optical glasses available to thedesigner are plotted on the (η₁,η₂) plane, suitable pairs of glassesthat could in principle be used for designing an optical system that iscolor-corrected at three wavelengths can be ascertained by drawingstraight lines from the origin of the coordinate system. Those glasseswhose glass points are collinear with each other along any straight linethrough the origin of the coordinate system are in principle suitablecandidates from which glass pairs can be selected for designing thetwo-glass three-color optical system.

Although any pair of glasses whose glass points are collinear with eachother and with the origin of the coordinate system as shown in FIG. 3could be used for designing a two-glass three-color optical system, itwould ordinarily be desirable to select a pair of glasses from amongthose glasses satisfying the further condition that the absolute valueof the difference between the corresponding primary dispersioncoefficients is large. The particular design form of the optical system,including a determination of the appropriate number of lens elements(e.g., doublet, triplet, or higher-number multiplet), is determined inaccordance with the principles of geometrical optics.

For the special case of a two-glass optical system that iscolor-corrected at four wavelengths (i.e., where n=4), the dispersionmodel used to derive the η-dispersion coefficients is cubic. As shown inFIG. 4, a two-glass four-color optical system can be designed using apair of optical glasses whose glass points are collinear with the originof the coordinate system defining the three-dimensional η-dispersioncoefficient space of the cubic dispersion model. Thus, if the glasspoints for the various optical glasses available to the designer areplotted in the three-dimensional (η₁,η₂,η₃) space, suitable pairs ofglasses that could in principle be used for designing an optical systemthat is color-corrected at four wavelengths can be ascertained bydrawing straight lines from the origin of the coordinate system. Thoseglasses whose glass points are collinear with each other along anystraight line through the origin of the coordinate system are inprinciple suitable candidates from which glass pairs can be selected fordesigning the two-glass four-color optical system. A preferred pair ofglasses would ordinarily comprise glasses for which the absolute valueof the difference between their corresponding primary dispersioncoefficients is as large as possible.

The degree of the dispersion model used to derive the η-dispersioncoefficients does not depend upon the number of glasses used in makingthe lens elements of the optical system. Therefore, the samethree-dimensional plot of glass points shown in FIG. 4 in connectionwith the design of a four-color optical system comprising two differenttypes of glasses could also be used for designing a four-color opticalsystem comprising three different types of glasses. A three-glass lenssystem must necessarily comprise at least three lens elements. For athree-glass optical system that is color-corrected at four wavelengths(i.e., where n=4), the dispersion model used to derive the η-dispersioncoefficients is cubic. As shown in FIG. 5, three glasses G₁, G₂ and G₃are suitable for designing an optical system that is color-corrected atfour or more wavelengths, if the glass points representing those threeglasses in three-dimensional η-dispersion coefficient space are coplanarwith each other and with the origin of the coordinate system.

In general, for an optical system that is color-corrected at nwavelengths, the dispersion model used to derive the η-dispersioncoefficients is an (n-1)th degree polynomial, where n is the number ofwavelengths for which color-correction is desired. Thus, a two-glassfive-color optical system could be designed using any pair of glasseswhose glass points lie on a common hyperline through the origin of anorthogonal coordinate system defining a four-dimensional hyperspace,where the axes of the coordinate system are the η-dispersioncoefficients according to a fourth degree dispersion model. Likewise, athree-glass five-color optical system could be designed using any threeglasses whose glass points lie on a common hyperplane that passesthrough the origin of the coordinate system defining the samefour-dimensional hyperspace.

In accordance with the present invention, an (n-1)th degree glassdispersion model is used to identify optical materials that arecompatible with each other for designing an optical systemcolor-corrected at n wavelengths. Each of the optical glasses availableto the designer is represented by a unique point in an (n-1)-dimensionalglass space, where the dispersion coefficients of the individual glassesare the coordinates of the glasses in the glass space. In practice, itis convenient to employ a digital computer to calculate and comparedispersion coefficient ratios for the various glasses available to thedesigner, where the dispersion coefficients are determined from a glassdispersion model in accordance with the number of wavelengths for whichcolor correction is required for the particular application. For mostcolor-correction requirements, it would not be necessary to use a glassdispersion model of order higher than cubic, i.e., n=4. For the analysisdescribed herein, the η-dispersion coefficients are used, although theν-dispersion coefficients could have been used.

In FIG. 6A, the glass points in η-dispersion coefficient space for 813different optical glasses listed in the current catalogs of the fiveleading optical glass manufacturers are plotted in the two-dimensionalη-dispersion coefficient space of the quadratic dispersion model. Thevalues of η₁ and η₂, which are called the primary and secondaryη-dispersion coefficients, respectively, for the glasses according to aquadratic dispersion model, were calculated from equation (7) using aleast-squares method for seven wavelengths in the visible region with abase wavelength of λ₀ =0.5740 μm. The seven wavelengths used were thesolar emission lines 0.4047 μm, 0.4350 μm, 0.4861 μm, 0.5461 μm, 0.5876μm, 0.6563 μm and 0.7065 μm.

The optical glasses whose glass point coordinates η₁ and η₂ are plottedgraphically in FIG. 6A are also listed in TABLE I. The 813 glassesmanufactured by the five leading optical glass manufacturers areidentified in TABLE I by manufacturer's name. i.e., SCHOTT, OHARA, HOYA,Corning-France (abbreviated as CORN-F) and Chance Pilkington(abbreviated as CHANCE), and by catalog number. Also given for eachglass listed in TABLE I are the base refractive index at λ₀ =0.574 μm,the primary dispersion coefficient η₁, the secondary dispersioncoefficient η₂, the ratio (η₁ /η₂), and the arc tangent of the ratio (η₁/η₂). The glasses are sorted in TABLE I in ascending order of the ratios(η₁ /η₂). The arc tangent of the ratio (η₁ /η₂) indicates the slopeangle relative to the η₁ -axis, so that glasses having substantially thesame arc tangent of the ratio (η₁ /η₂) are substantially collinear, andare therefore candidates for selection in forming pairs of glasses fordesigning optical systems that are color-corrected at three or morewavelengths. FIG. 6B provides an enlargement of a portion of the plotshown in FIG. 6A.

The glass selection procedure in accordance with the present inventioncan be illustrated graphically with reference to FIGS. 6A and 6B. Inselecting a pair of glasses from among all the optical glasses whoseglass points are plotted in FIG. 6A for designing a two-glass opticalsystem that is color-corrected at three wavelengths, a designer firstchooses one glass that is available to him and locates the glass pointrepresenting that one glass on the plot. The designer then draws astraight line extending through the glass point for that one glass andthrough the origin of the coordinate system. Other glasses that inprinciple are suitable for use in combination with that one glass fordesigning the desired two-glass three-color lens doublet liesubstantially on that same straight line.

All glass points on the same straight line through the origin of thecoordinate system have the same slope angle relative to the η₁ -axis.Therefore, a tabulation of the available optical glasses in terms of aconvenient function of the slope angle, e.g., the arc tangent of theratio (η₁ /η₂), as provided in Table I, can be used by an opticaldesigner with greater convenience and accuracy than the graphicaltechnique described above. With such a tabulation, all that the designerneed do in selecting a pair of glasses for designing a two-glassthree-color lens doublet is to choose a first glass, and then toidentify other glasses whose tabulated values for the arc tangent of theratio (η₁ /η₂) are substantially the same as for the chosen first glass.A suitable pair of glasses can be selected from among the glasses havingsubstantially the same value for the arc tangent of the ratio (η₁ /η₂).

Each of the glasses listed in TABLE I can be identified unambiguously bythe manufacturer's catalog number, which might be an alphanumericdesignation or a purely numeric designation depending upon themanufacturer. In a case where a new type of glass is manufactured bychanging the constituents or percentage composition of an older type ofglass, the new type of glass is invariably given a new catalog number bythe manufacturer. Even where the new type of glass is made to meet thesame optical specifications as an older type of glass, the new type ofglass is given a new catalog number. The process of making anyparticular type of optical glass is ordinarily regarded by themanufacturer as proprietary information, and is ordinarily not known indetail by the optical designer who uses the glass. Accordingly, opticaldesigners universally identify particular optical glasses bymanufacturers' catalog numbers.

Considering the portion of the plot of FIG. 6A shown in enlarged view inFIG. 6B, it is instructive to identify glasses that could be pairedwith, for example, Hoya FCD10 glass for designing a two-glass opticalsystem that is color-corrected at three wavelengths. Hoya FCD10 glass,with coordinates η₁ =-0.0610293 and η₂ =-0.00399528 in two-dimensionalη-dispersion coordinate space (as ascertained by reference to page 10 ofTABLE I), was selected for instructional purposes because the glasspoint for this particular optical glass is relatively isolated on theplot in FIG. 6B. A straight line, Line A, drawn through the origin ofthe coordinate system and the glass point for Hoya FCD10 glass indicatesthat, e.g., Hoya FC3 glass (coordinates η₁ =-0.0842094 and η₂=-0.00547060) can be combined with Hoya FCD10 glass for designing atwo-glass optical system that is color-corrected at at least threewavelengths. An examination of the cubic and quartic dispersion modelsfor these two glasses in the same spectral region as covered by TABLE Ifor the quadratic dispersion model shows that this particular pair ofoptical glasses can yield axial color correction at up to fivewavelengths. This phenomenal compatibility of the Hoya FCD10 and HoyaFC3 glasses is discussed hereinafter in connection with a Mangin doubletmirror as illustrated in FIG. 21.

Line A in FIG. 6B also shows several other glasses that could becombined with Hoya FCD10 glass to form pairs of compatible glasses fordesigning optical systems that are color-corrected at at least threewavelengths. Thus, Schott BK8 glass (coordinates η₁ =-0.0870925 and η₂=-0.00577220), Schott PSK50 glass (coordinates η₁ =-0.0823975 and η₂=-0.00530444), Hoya PCD2 glass (coordinates η₁ =-0.0878957 and η₂=-0.00577861), Hoya ZNC7 glass (coordinates η₁ =-0.0905172 and η₂=-0.00582287), and Chance Pilkington 641601 glass (coordinates η₁=-0.0923210 and η₂ =-0.00588984) are also all substantially collinearwith each other and with Hoya FCD10 glass and Hoya FC3 glass on Line A,and could in principle be considered along with Hoya FCD10 glass andHoya FC3 glass as suitable candidates from which to select a pair ofglasses for designing the desired two-glass three-color optical system.For optimum dioptric power distribution, however, it would usually bedesirable to select a pair of glasses whose glass points are widelyseparated from each other along Line A, so that the difference betweenthe η₁ primary dispersion coefficients for the two glasses is large.

As another example, it is instructive to consider the glasses that couldbe combined with Schott PK51 glass (coordinates η₁ =-0.0718368 and η₂=-0.00220267) for designing a two-glass optical system that iscolor-corrected at at least three wavelengths. A line, Line B, drawnthrough the origin of the coordinate system and the glass point forSchott PK51 glass indicates that, e.g., Hoya LAC7 glass (coordinates η₁=-0.0946047 and η₂ =-0.00290729 could be paired with Schott PK51 glassto form a pair of compatible glasses for designing an optical systemthat is color-corrected at at least three wavelengths. An examination ofthe cubic and quartic dispersion models for these two glasses in thesame spectral region as covered by TABLE I for the quadratic dispersionmodel shows that this particular pair of glasses can be used fordesigning opticl systems that are color-corrected at five wavelengths.This phenomenal compatibility of the Schott PK51 and Hoya LAC7 glassesis demonstrated hereinafter by the lens doublet shown in FIG. 12, thelens quadruplet shown in FIG. 19, the Mangin doublet mirror shown inFIG. 24, the Petzval lens system shown in FIG. 27, and the prism systemsshown in FIGS. 30, 31 and 32.

A glance at Line B indicates that a number of other glasses, whose glasspoints lie substantially on Line B, are also suitable candidates for usein combination with Schott PK51 glass for designing an optical systemthat is color-corrected at three wavelengths, e.g., Schott K11 glass(coordinates η₁ =-0.0900976 and η₂ =-0.00275514), Schott LAK21 glass(coordinates η₁ =-0.0920718 and η₂ =-0.00285003), and Corning-France511604 glass (coordinates η₁ =-0.0916603 and η₂ =-0.00283942).

It is important to remember that the values of η₁ and η₂ for the opticalglasses listed in TABLE I were calculated using a least-squares methodfor seven selected wavelengths in the visible spectrum extending from aminimum wavelength of approximately 0.4 μm to a maximum wavelength ofapproximately 0.7 μm, with a base wavelength of 0.5740 μm. Thesewavelengths provide a statistically good representation of the entirevisible spectrum. However, in a particular application, the designermight find it preferable to calculate values of η₁ and η₂ for theoptical materials of interest by a least-square method, or indeed by anyother mathematical procedure, using different wavelengths. For example,if the designer were concerned with color-correction for infraredwavelengths, he could calculate a table of values for η₁ and η₂ forseveral wavelengths in the infrared length band or bands of interest. Ifthe designer were interested in obtaining color-correction at a numberof different wavelength bands, as might be appropriate in designing,e.g., a color-corrected astronomical instrument having a common focalplane for radiation in the infrared and ultraviolet portions of thespectrum, he could calculate a table of values for η₁ and η₂ usingseveral wavelengths in each of the wavelength bands of interest.

The values calculated for η₁ and η₂, and therefore also the valuescalculated for the ratios (η₁ /η₂) and for the arc tangents of theratios (η₁ /η₂), depend upon the wavelengths used in the quadraticdispersion model obtained from equation (7). In general, therefore, thevalues for η₁ and η₂ would differ from those shown in TABLE I, ifdifferent wavelengths were to be used in representing the spectral range(or the discontinuous spectral bands) of interest to the designer. Thevalues listed in TABLE I, however, are useful in illustrating theinvention, and furthermore are of practical utility in selecting pairsof compatible glasses for designing optical systems that aresubstantially color-corrected at at least three wavelengths in thevisible spectrum.

In TABLE II, the same glasses manufactured by the same five opticalglass manufacturers are again listed and unambiguously identified bymanufacturer's name and catalog number, along with the base refractiveindex at λ₀ =0.589 μm for each glass. Also listed for each glass inTABLE II are the primary dispersion coefficient η₁, the secondarydispersion coefficient η₂, the tertiary dispersion coefficient η₃, andthe ratios (η₁ /η₂), (η₂ /η₃) and (η₁ /η₃), as calculated from equation(7) using a least-squares fitted cubic dispersion model. For each glass,the η-dispersion coefficients η₁, η₂ and η₃, as well as the baserefractive index, were obtained for ten wavelengths in the extendedspectral region from 0.3650 μm to 1.014 μm, with a base wavelength of λ₀=0.589 μm. The glasses are sorted in TABLE II in ascending order of theratios (η₁ /η₂). The ten wavelengths used were the solar emission lines0.3650 μm, 0.4047 μm, 0.4350 μm, 0.4861 μm, 0.5461 μm, 0.5876 μm, 0.6563μm, 0.7065 μm, 0.7682 μm and 1.014 μm.

Ordinarily, those optical glasses for which the corresponding ratios (η₁/η₂), (η₂ /η₃) and (η₁ /η₃) are substantially equal to each other, andfor which the glass points are substantially collinear with each otherand with the origin of the coordinate system, are candidates forselection in forming pairs of compatible glasses to use for designingoptical systems that are color-corrected at four wavelengths. It issometimes more convenient to use the arc tangents of the ratios (η₁/η₂), (η₂ /η₃) and (η₁ /η₃) rather than the ratios themselves inselecting compatible pairs of glasses for designing optical systems thatare color-corrected at at least four wavelengths.

Since the η-dispersion coefficients and the ν-dispersion coefficientsfor any optical material are related by the equation η_(ij) =ν_(ij) /(N₀-1), the selection of glass pairs for designing a two-glass three-coloroptical system could also be made from a plot of the glass points inν-dispersion coefficient space, or from a tabulation of the values forthe arc tangents of the ratios (ν₁ /ν₂), for the various optical glassesavailable to the designer. In FIG. 7A, the glass points are plotted inν-dispersion coefficient space for the same 813 glasses whose glasspoints in η-dispersion coefficient space are plotted in FIG. 6A. Anenlargement of a portion of the plot shown in FIG. 7A is shown in FIG.7B.

To illustrate graphically the method of selecting a pair of compatibleglasses for designing a two-glass three-color optical system using theν-dispersion coefficients for the glasses available to the designer,consider Schott FK51 glass as the initial selection. Schott FK51 glassis represented in FIG. 7B by a glass point whose coordinates inν-dispersion coefficient space are ν₁ =-0.031926 and ν₂ =-0.001819, asdetermined by the quadratic dispersion model used to plot the glasspoints shown in FIGS. 7A and 7B. A straight line, Line C, drawn throughthe origin of the coordinate system and the glass point for Schott FK51glass indicates a number of other optical glasses whose glass points aresubstantially collinear with the origin and with the glass point forSchott FK51 glass. Examples of other suitable glasses lying on Line Cinclude Schott K51 glass and Hoya ADF1 glass. All such glasses, whoseglass points are substantially collinear with each other and with theorigin, are suitable candidates from which to select a pair of glassesfor designing an optical system having the required color correction atthree wavelengths.

In FIG. 7B, straight lines could be drawn at various angles from theorigin through the glass points representing the optical glassesavailable to the designer, thereby identifying suitable pairs of glassesfor designing two-glass three-color optical systems. Thus, Line D inFIG. 7B indicates that, e.g., Corning-France 558525 glass and SchottBAF50 glass can be paired for designing a two-glass three-color opticalsystem.

In FIG. 6A or FIG. 7A, any pair of optical glasses whose glass pointslie substantially on the same straight line passing through the originof the coordinate system can in principle be used for designing atwo-glass three-color lens systems. Conversely, optical glasses whoseglass points do not lie substantially on the same straight line cannotbe used for designing a two-glass lens system that is color-corrected atthree wavelengths. From equations (41) to (46), the dioptric powers φ₁and φ₂ of the individual lens elements of a two-glass three-color lensdoublet are inversely proportional to the absolute value of thedifference between the primary dispersion coefficients η₁₁ and η₁₂ ofthe glasses used to make the lens elements of the doublet. It istherefore necessary that the absolute value |η₁₂ -η₁₁ | be as large aspossible in order to minimize the dioptric powers of the individual lenselements. If the dioptric powers of the lens elements were too large,the configurations of the lens elements might be so rounded (resemblingtoy marbles in an extreme case) that it would be impractical to use themin most optical systems. The requirement that |η₁₂ -η₁₁ | be largewould, therefore, ordinarily be treated as an additional condition thatmust be satisfied when selecting a pair of glasses for designing acolor-corrected optical system.

In FIG. 6B or 7B, those optical glasses whose glass points aresubstantially collinear with each other and with the origin of thecoordinate system are suitable candidates from which to select a pair ofglasses for designing a two-glass optical system that is color-correctedat three or more wavelengths. Those glasses whose glass points aresubstantially collinear with the origin have ratios of primary tosecondary dispersion coefficients that are substantially equal. Thus, inmathematical terms, when

    η.sub.11 /η.sub.21 =η.sub.12 /η.sub.22,

or, equivalently, when

    ν.sub.11 /ν.sub.21 =ν.sub.12 /ν.sub.22,

for two optical glasses, those two glasses in combination are suitablefor designing a two-glass optical system that is color-corrected atthree wavelengths. This property can readily be exploited with a digitalcomputer by computing the slope of the line connecting each glass pointwith the origin, and then sorting all the glasses in ascending order ofthese "glass point" slopes.

It was known in the prior art that any two optical glasses can inprinciple be used in combination with each other for designing a lensdoublet that is color-corrected at two wavelengths. This prior artknowledge is seen to be a special case of the unified theory upon whichthe present invention is based, where n=2. In FIG. 8, a cemented lensdoublet of unit focal length at λ₀ =0.54607 μm for a relative apertureof f/10 is shown, where the lens elements comprising the doublet aremade of Schott BK7 glass and Schott F2 glass. The two lens elemets couldbe cemented together by any technique well-known in the art, as by usingCanada balsam, optical cement or immersion oil. It can be seen byreference to page 9 of TABLE I that the ratio η₁ /η₂ =-13.6653 forSchott BK7 glass, and by reference to page 7 of TABLE I that the ratioη₁ /η₂ =-3.91263 for Schott F2 glass, are quite different in value.Thus, the unified theory upon which the present invention is basedindicates that a lens doublet made of Schott BK7 and Schott F2 glassescannot provide color correction at more than two wavelengths. Colorcorrection is a function of the choice of glasses used for designing thedoublet, and is independent of the particular design form of the doubletmade from the chosen glasses. Thus, no amount of manipulation of thedesign form of a lens doublet made from these two glasses can result incolor correction at more than two wavelengths.

It is a feature of the present invention that the suitability ornon-suitability of any particular combination of optical materials fordesigning a color-corrected optical system can be readily determined byreference to tabulated data, such as by compairing tabulated values ofthe ratios η₁ /η₂ for a pair of optical materials being considered. Thepresent invention eliminates the need for "cut-and-try" methods fordetermining whether any particular pair of optical materials can providecolor correction at more than two wavelengths.

Although any two optical glasses are suitable in principle for designinga two-glass two-color lens doublet, there was no unified theory in theprior art for predicting whether any two particular optical glasseswould have appropriate dioptric powers so that lens elements withpractical configurations could be made from those particular glasses. Itis a feature of the present invention, on the other hand, that thepracticality or non-practicality of any two particular optical glassesfor designing a lens doublet that is color-corrected at two or morewavelengths can be readily determined by reference to tabulated data.

The cemented lens doublet shown in FIG. 8, comprising lens elements madeof Schott BK7 and Schott F2 glasses, is color-corrected at only twowavelengths. By reference to TABLE I, it can be ascertained that theabsolute value of the difference between corresponding values of η₁ forSchott BK7 and Schott F2 glasses is quite large, i.e., |η₁₂ -η₁₁|=0.063435, indicating that the distribution of dioptric powers for lenselements made from these two glasses would be desirably small. Thus, thepresent invention provides a method for determining the practicality ofusing two particular glasses, e.g., Schott BK7 glass and Schott F2glass, in combination for designing a two-glass two-color lens doublet.As an empirical rule, it has been found that where the absolute value ofthe difference between the values of the primary dispersion coefficientsη_(1j) (where j=1, 2) for two optical glasses is greater than about0.02, i.e., when |Δη_(1j) |=|η₁₂ -η₁₁ | >0.02, the dioptric powers oflens elements made from those two glasses will be small enough so thatthe lens elements will have sufficiently large radii of curvature toenable a lens doublet of practical configuration to be made.

Also shown in FIG. 8 is a curve showing the relationship of paraxialmarginal ray height (conventionally designated PY in the drawing) towavelength for the lens doublet comprising first and second lenselements made of Schott BK7 and Schott F2 glasses, respectively. Such acurve, hereinafter called a "wavelength scan", for an optical systemprovides an indication of the quality of axial color correctionobtainable with the optical system. The paraxial marginal ray height foran optical system at any particular wavelength is a measure of theso-called secondary spectrum of the system at that wavelength, i.e., ameasure of the extent to which light at that particular wavelength isout-of-focus with respect to the common focus for which color-correctionof the system is desired. Perfect color correction is achieved for thosewavelengths at which the paraxial marginal ray height is zero, i.e, forthose wavelengths at which the "wavelength scan" curve crosses thehorizontal axis. For wavelengths at which perfect color correction isnot achieved, the closeness of the "wavelength scan" curve to thehorizontal axis provides a measure of the extent to which the opticalsystem approaches perfect color correction.

For the lens doublet shown in FIG. 8, color correction is obtained atone wavelength in the blue portion of the visible spectrum and at onewavelength in the infrared spectrum. In between these twocolor-corrected wavelengths, however, the dispersion produced by thelens elements of the doublet is quite pronounced. The "wavelength scan"curve in FIG. 8 indicates that a lens doublet made Schott BK7 and SchottF2 glasses cannot provide a good degree of color correction atwavelengths between the crossings of the horizontal axis, because thecurve (i.e., the paraxial marginal ray height) departs rapidly from thehorizontal axis even for wavelengths close to the two crossings of thehorizontal axis.

INDUSTRIAL APPLICABILITY Two-Glass Lens Doublets

FIG. 9 illustrates a particular example of a three-color cemented lensdoublet in which the optical glasses from which the two lens elements ofthe doublet are made were selected in accordance with the presentinvention. The lens doublet of FIG. 9 is of unit focal length at λ₀=0.54607 μm for a relative aperture of f/10, and the two glasses used inmaking the lens elements of the doublet are Schott FK51 and Schott K51glasses. These glasses were chosen because the ratios of their primaryto secondary η-dispersion coefficients are substantially equal, andbecause the absolute value of the difference between correspondingvalues of the primary dispersion coefficient η₁ for the two glasses islarge.

The wavelength scan curve shown in FIG. 9 crosses the horizontal axis atthree discrete wavelengths in the visible spectrum, and is close to thehorizontal axis throughout the entire wavelength band extending over thevisible spectrum. Thus, a lens doublet made from this particular pair ofoptical glasses can provide very effective (if not perfect) colorcorrection over substantially the entire visible spectrum.

The design form of the lens doublet shown in FIG. 9 is specified intabular format as follows:

    ______________________________________                                        Surface                                                                       No.     Radius   Thickness N.sub.e                                                                             V.sub.e                                                                            Material                                ______________________________________                                        1        .421 f  .016 f    1.48794                                                                             84.07                                                                              Schott FK51                             2       -.215 f  .009 f    1.50720                                                                             59.34                                                                              Schott K51                              3        7.034 f                                                              ______________________________________                                    

where the various lens element surfaces of the doublet system arenumbered, in accordance with conventional practice, from left to right.The "Radius" listed for each surface is the radius of curvature of thesurface expressed as a multiple of the focal length f of the lens systemat a relative aperture of f/10. In accordance with convention, theradius of curvature of an optical surface is said to be positive if thecenter of curvature of the surface lies to the right of the surface, andnegative if the center of curvature of the surface lies to the left ofthe surface. The "Thickness" listed for a particular surface is thethickness of the lens element bounded on the left by the indicatedsurface, where the thickness is measured along the optical axis of thelens system. The thickness of each lens element is expressed as amultiple of the focal length f of the lens system. N_(e) is therefractive index of the lens element bounded on the left by theindicated surface, where the value of the refractive index is given forthe mercury e line, i.e., for a wavelength of λ₀ =0.54607 μm. V_(e) isthe Abbe number for the same lens element at the same base wavelength."Material" refers to the type of optical material from which the lenselement is made, where the type of optical material is identified bymanufacturer and by manufacturer's catalog number.

FIG. 10 illustrates a particular example of a four-color cemented lensdoublet in which the optical glasses from which the two lens elements ofthe doublet are made were selected in accordance with the presentinvention. The lens doublet of FIG. 10 is of unit focal length at λ₀=0.54607 μm for a relative aperture of f/10. The glasses used in makingthe two lens elements of the doublet of FIG. 10 are Corning France486817 and Ohara LAK14 glasses, which were chosen because their ratios(η₁ /η₂) are substantially equal in the extended spectral region from0.365 μm to 1.014 μm, and because the difference between correspondingvalues of η₁ for the two glasses is large.

The wavelength scan curve shown in FIG. 10 crosses the horizontal axisat four discrete wavelengths in the visible and near infrared spectrum,and is close to the horizontal axis throughout this spectral range.Thus, a lens doublet made from this particular pair of optical glassescan provide very effective color correction over a wide spectral rangebridging the visible and infrared wavelengths, especially in the rangeextending from the blue wavelengths through the near infraredwavelengths beyond 1.0 μm.

The design form of the lens doublet shown in FIG. 10 is specified intabular format as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1       .257 f  .016 f    1.48746                                                                             81.40                                                                              Corn-F 486817                            2      -.372 f  .009 f    1.69979                                                                             55.31                                                                              Ohara LAK14                              3       2.022 f                                                               ______________________________________                                    

where the specified design parameters are as defined above in connectionwith FIG. 9.

FIG. 11 illustrates a particular example of a four-color air-spaced lensdoublet in which the two lens elements are made of the same types ofoptical glasses as for the cemented lens doublet shown in FIG. 10. Thelens doublet of FIG. 11 is likewise of unit focal length at λ₀ =0.54607μm for a relative aperture of f/10. The wavelength scan curve shown inFIG. 11 crosses the horizontal axis at four discrete wavelengths in thevisible and near infrared spectrum, and is close to the horizontal axisthroughout the same spectral range for which the lens doublet of FIG. 10provides effective color correction. Thus, the degree of colorcorrection obtainable using the glasses of the cemented lens doubletshown in FIG. 10 is retained when an air space is provided between thelens elements, which implies that the quality of axial (i.e.,longitudinal) color correction obtainable with a two-glass lens doubletis determined by the choice of glasses used for making the lenselements, and is independent of the design form of the doublet. Thequality of correction for lateral chromatic abberation, spherochromatismand the remaining aberrations, however, depends upon the design form ofthe lens doublet.

The design form of the lens doublet shown in FIG. 11 is specified intabular format as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1       .598 f  .016 f    1.48746                                                                             81.40                                                                              Corn-F 486817                            2      -.207 f   .0005 f             Air                                      3      -.221 f  .009 f    1.69979                                                                             55.31                                                                              Ohara LAK14                              4      -.699 f                                                                ______________________________________                                    

FIG. 12 illustrates a particular example of a five-color cemented lensdoublet in which the optical glasses from which the two lens elements ofthe doublet are made were selected in accordance with the presentinvention. The lens system of FIG. 12 is of unit focal length at λ₀=0.54607 μm for a relative aperture of f/10. The glasses used in makingthe two lens elements of the cemented doublet of FIG. 12 are Schott PK51and Hoya LAC7 glasses, which were chosen because the ratios of theirprimary to secondary η-dispersion coefficients are substantially equal,and because the absolute value of the difference between correspondingvalues of η₁ for the two glasses is large.

The wavelength scan curve shown in FIG. 12 crosses the horizontal axisat five discrete wavelengths in the spectral range extending from theultraviolet to the near infrared wavelengths. Furthermore, thewavelength scan curve is extremely close to the horizontal axisthroughout the spectral range extending from the green wavelengths toabout 0.8 μm in the near infrared region. Thus, a lens doublet made fromthis particular pair of optical glasses can provide very effective axialcolor correction throughout this region of the optical spectrum.

The design form of the lens doublet shown in FIG. 12 is specified intabular format as follows:

    ______________________________________                                        Surface                                                                       No.     Radius   Thickness N.sub.e                                                                             V.sub.e                                                                            Material                                ______________________________________                                        1        .753 f  .020 f    1.53019                                                                             76.57                                                                              Schott PK51                             2       -.151 f   .0004 f             Air                                     3       -.156 f  .010 f    1.65426                                                                             58.15                                                                              Hoya LAC7                               4       -.680 f                                                               ______________________________________                                    

Two-Glass Lens Triplets

FIG. 13 illustrates an example of a four-color cemented two-glass lenstriplet in which the optical glasses from which the three lens elementsof the triplet are made were selected in accordance with the presentinvention. The lens triplet as shown in FIG. 13 has a focal length thatis 40 times the focal length of the lens doublets shown in FIGS. 8-12 atλ₀ =0.54607 μm for a relative aperture of f/10. The first and third lenselements of the lens triplet of FIG. 13 are made of Corning-France486817 glass, and the second (i.e., middle) lens element is made ofCorning-France 589612 glass. These glass types were chosen because theratios of their primary to secondary η-dispersion coefficients aresubstantially equal, and because the absolute value of the differencebetween corresponding values of η₁ for the two glasses is large.

The wavelength scan curve shown in FIG. 13 crosses the horizontal axisat four discrete wavelengths in the spectral range extending from theblue to the near infrared wavelengths, and is close to the horizontalaxis throughout this spectral range. Thus, a lens triplet made from thisparticular pair of optical glasses can provide very effective axialcolor correction throughout this region of the optical spectrum.

It is convenient to specify the design form of the lens triplet of FIG.13 in terms of unit focal length for a relative aperture of f/10, as wasdone above for the lens doublets of FIGS. 8-12. It is noted, however,that the depiction in FIG. 13 shows a lens triplet whose focal length is40 times that of the lens doublets shown in FIGS. 8-12. In terms of thefocal length f, the design form of the lens triplet of FIG. 13 isspecified in the tabular format as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1        .368 f .015 f    1.48746                                                                             81.40                                                                              Corn-F 486817                            2      -2.029 f .009 f    1.59129                                                                             60.94                                                                              Corn-F 589612                            3        .212 f .013 f    1.48746                                                                             81.40                                                                              Corn-F 486817                            4      -2.324 f                                                               ______________________________________                                    

FIG. 14 illustrates a particular example of a four-color two-glass lenstriplet in which the three lens elements are made of the same types ofoptical glasses as for the cemented lens triplet shown in FIG. 13, withthe first and second lens elements being cemented together, and with anair gap being provided between the second and third lens elements. Thefocal length of the lens triplet shown in FIG. 14 is the same as for thelens triplet shown in FIG. 13 for the same relative aperture of f/10.

The wavelength scan curve shown in FIG. 14 likewise crosses thehorizontal axis at four discrete wavelengths in the spectral rangeextending from the blue to the near infrared wavelengths, and is closeto the horizontal axis throughout this spectral range. Thus, the degreeof axial color correction obtainable using the pair of glasses of thecemented lens triplet shown in FIG. 13 is seen to be retained when anair space is provided between two of the lens elements. The wavelengthscan thus indicates that, for a given pair of optical materials, thequality of axial color correction obtainable with a two-glass lenstriplet is determined by the choice of glasses used to make the lenselements of the triplet, and is independent of the design form of thetriplet. Of course, the quality of correction for lateral coloraberration, spherochromatism and the higher-order monochromaticaberrations is dependent upon the design form of the lens triplet.

In terms of the focal length, the design form of the lens triplet ofFIG. 14 is specified in tabular format as follows:

    ______________________________________                                        Surface           Thick-                                                      No.    Radius     ness    N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1      .419 f     .015 f  1.48746                                                                             81.40                                                                              Corn-F 486817                            2      -.449 f    .009 f  1.59129                                                                             60.94                                                                              Corn-F 589612                            3      .342 f     .001 f             Air                                      4      .360 f     .013 f  1.48746                                                                             81.40                                                                              Corn-F 486817                            5      -1.101 f                                                               ______________________________________                                    

Two-Glass Lens Quadruplets

FIG. 15 illustrates a particular example of a four-color cementedtwo-glass lens quadruplet having a focal length that is 40 times thefocal length of the lens doublets shown in FIGS. 8-12 at λ₀ =0.54607 μmfor a relative aperture of f/10. The optical glasses from which the fourlens elements of the quadruplet shown in FIG. 15 are made were selectedin accordance with the present invention. The first and third lenselements are made of Corning-France 486817 glass, and the second andfourth lens elements are made of Corning-France 589612 glass. Theseglasses were chosen because the ratios of their primary to secondaryη-dispersion coefficients are substantially equal, and because theabsolute value of the difference between corresponding values of η₁ forthe two glasses is large.

The wavelength scan curve shown in FIG. 15 crosses the horizontal axisat four discrete wavelengths in the spectral range extending from theblue to the near infrared region of the optical spectrum. The wavelengthscan curve is close to the horizontal axis throughout this spectralrange, which indicates that a lens quadruplet made from this particularpair of glasses can provide effective axial color correction throughoutthis spectral range.

In terms of the focal length, the design form of the lens quadruplet ofFIG. 15 is specified in tabular format as follows:

    ______________________________________                                        Surface           Thick-                                                      No.    Radius     ness    N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1      .437 f     .015 f  1.48746                                                                             81.40                                                                              Corn-F 486817                            2      5.098 f    .009 f  1.59129                                                                             60.94                                                                              Corn-F 589612                            3      .285 f     .015 f  1.48746                                                                             81.40                                                                              Corn-F 486817                            4      -.363 f    .009 f  1.59129                                                                             60.94                                                                              Corn-F 589612                            5      -1.147 f                                                               ______________________________________                                    

FIG. 16 illustrates a particular example of a four-color two-glass lensquadruplet in which the four lens elements are made of the same types ofoptical glasses as for the cemented lens quadruplet shown in FIG. 15,with the first and second lens elements being cemented together, thethird and fourth lens elements being cemented together, and an air gapbeing provided between the second and third lens elements. The focallength of the lens quadruplet shown in FIG. 16 is the same as for thelens quadruplet shown in FIG. 15 for the same relative aperture of f/10.

Th wavelength scan curve shown in FIG. 16 likewise crosses thehorizontal axis at four discrete wavelengths in the spectral rangeextending from the blue to the near infrared region of the opticalspectrum. The wavelength scan curve of FIG. 16 is likewise close to thehorizontal axis throughout this spectral range. As can be seen bycomparing the wavelength scan of FIG. 16 with the wavelength scan ofFIG. 15, the degree of axial color correction obtainable using the pairof glasses of the cemented lens quadruplet of FIG. 15 is retained whenan air space is provided between two of the lens elements.

In terms of the focal length, the design form of the lens quadruplet ofFIG. 16 is specified in tabular format as follows:

    ______________________________________                                        Surface           Thick-                                                      No.    Radius     ness    N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1      .948 f     .015 f  1.48746                                                                             81.40                                                                              Corn-F 486817                            2      -.375 f    .009 f  1.59129                                                                             60.94                                                                              Corn-F 589612                            3      1.280 f    .001 f             Air                                      4      .499 f     .009 f  1.59129                                                                             60.94                                                                              Corn-F 589612                            5      .274 f     .015 f  1.48746                                                                             81.40                                                                              Corn-F 486817                            6      -1.150 f                                                               ______________________________________                                    

FIG. 17 illustrates a particular example of a four-color two-glass lensquadruplet in which the four lens elements are made of the same types ofoptical glasses as for the lens quadruplets shown in FIGS. 15 and 16,except that the first and third lens elements are made of Corning-France486817 glass and the second and fourth lens elements are made ofCorning-France 589612 glass. The focal length of the lens quadrupletshown in FIG. 17 is the same as for the lens quadruplets shown in FIGS.15 and 16 for the same relative aperture of f/10. As with the lensquadruplet shown in FIG. 16, the first and second lens elements of thelens quadruplet shown in FIG. 17 are cemented together, the third andfourth lens elements are cemented together, and an air gap is providedbetween the second and third lens elements.

The wavelength scan curve shown in FIG. 17 likewise crosses thehorizontal axis at four discrete wavelengths in the spectral rangeextending from the blue to the near infrared region of the opticalspectrum. The wavelength scan curve of FIG. 16 is likewise close to thehorizontal axis throughout this spectral range. When compared with thewavelength scans shown in FIGS. 15 and 16, the wavelength scan shown inFIG. 17 further indicates that the quality of axial color correctionobtainable with a lens multiplet is determined by the choice of glassesused for designing the multiplet, and is independent of the design formof the multiplet.

In terms of the focal length, the design form of the lens quadruplet ofFIG. 17 is specified in tabular format as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1       .823 f  .015 f    1.48746                                                                             81.40                                                                              Corn-F 486817                            2      -.400 f  .009 f    1.59129                                                                             60.94                                                                              Corn-F 589612                            3       1.069 f .001 f               Air                                      4       .645 f  .015 f    1.48746                                                                             81.40                                                                              Corn-F 486817                            5      -.337 f  .009 f    1.59129                                                                             60.94                                                                              Corn-F 589612                            6      -.758 f                                                                ______________________________________                                    

Two-Glass Collimators

FIG. 18 shows a four-color collimator comprising a two-glass lensquadruplet, which can be used for precise collimation of a beam of lighthaving a spectral range from blue to near infrared. The collimator ofFIG. 18 comprises a lens quadruplet wherein the consecutive lenselements are all air-spaced. The first and fourth lens elements of thecollimator are made of Hoya FCD10 glass, and the second and third lenselements are made of Ohara LAK14 glass.

The wavelength scan curve shown in FIG. 18 crosses the horizontal axisat four discrete wavelengths, and is close to the horizontal axisthroughout a wavelength band that bridges the visible and near infraredspectral regions. In fact, the wavelength scan curve of FIG. 18substantially coincides with the horizontal axis throughout the nearinfrared spectral range, which illustrates the suitability of thisparticular combination of optical glasses for correcting paraxialchromatic aberration in the near infrared spectral range.

The collimator shown in FIG. 18, in addition to being substantiallycolor-corrected in the near infrared spectral range, is alsowell-corrected for the various monochromatic aberrations. In terms ofthe focal length for a relative aperture of f/10, the design form of thecollimator of FIG. 18 is specified in detail in tabular format asfollows:

    ______________________________________                                        Surface          Thick-                                                       No.    Radius    ness     N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1       1.244 f  .015 f   1.45770                                                                             90.32                                                                              Hoya FCD10                               2      -.381 f   .010 f              Air                                      3      -.381 f   .009 f   1.69979                                                                             55.31                                                                              Ohara LAK14                              4      -.542 f   .002 f              Air                                      5      -.539 f   .009 f   1.69979                                                                             55.31                                                                              Ohara LAK14                              6      -3.845 f   .0005 f            Air                                      7       .715 f   .015 f   1.45770                                                                             90.32                                                                              Hoya FCD10                               8      -1.025 f                                                               ______________________________________                                    

FIG. 19 shows a five-color collimator comprising a two-glass lensquadruplet. The first and fourth lens elements of the collimator of FIG.19 are made of Schott PK51 glass, and the second and third lens elementsare made of Hoya LAC7 glass. These two optical glasses were selectedbecause of the substantial collinearity of their respective glass pointswith the origin of the coordinate system in four-dimensionalη-dispersion coefficient space, and also because the absolute value ofthe difference between their respective primary dispersion coefficientsis large.

The wavelength scan curve shown in FIG. 19 crosses the horizontal axisat five discrete wavelengths in the spectral range extending from about0.38 μm to about 0.80 μm. These five wavelengths can therefore beprecisely focussed using the collimator of FIG. 19. The otherwavelengths in this spectral range can be focussed with a degree ofprecision indicated by the closeness with which the wavelength scancurve approaches the horizontal axis.

The collimator shown in FIG. 19 is well-corrected for the variousmonochromatic aberrations, especially spherochromatism. In terms of thefocal length for a relative aperture of f/10, the design form of thecollimator of FIG. 19 is specified in tabular format as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                            Material                                ______________________________________                                        1      -1.314  f    .018 f  1.53019                                                                              76.57                                                                              Schott PK51                           2      -.295   f    .025 f              Air                                   3      -.269   f    .010 f  1.65426                                                                              58.15                                                                              Hoya LAC7                             4      -1.795  f    .003 f              Air                                   5      .469    f    .010 f  1.65426                                                                              58.15                                                                              Hoya LAC7                             6      .233    f    .002 f              Air                                   7      .232    f    .015 f  1.53019                                                                              76.57                                                                              Schott PK51                           8      -.697   f                                                              ______________________________________                                    

FIG. 20 shows a five-color collimator comprising a lens quadruplethaving a focal length that is 40 times the focal length of the doubletsshown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/8.The first and fourth lens elements of the collimator of FIG. 20 are madeof calcium fluoride crystal, and the second and third lens elements aremade of Hoya LAC7 glass. Use of calcium fluoride crystal as one of theoptical materials enables a larger aperture (i.e., a lower f-number) tobe used than would be possible if both optical materials were glasses.With the collimator of FIG. 20, a light beam of large diameter can becollimated without significant error due to spherochromatism. Thewavelength scan curve shown in FIG. 20 crosses the horizontal axis atfive wavelengths, and therefore indicates that precise collimation canbe obtained as five wavelengths, in the spectral range from ultravioletto near infrared. The closeness of the wavelength scan curve to thehorizontal axis throughout this spectral range indicates that a highdegree of collimation can be obtained for the other wavelengths in thisrange.

In terms of the focal length for a relative aperture of f/8, the designform of the collimator of FIG. 20 is specified in tabular format asfollows:

    ______________________________________                                        Surface         Thick-                                                        No.    Radius   ness    N.sub.e                                                                              V.sub.e                                                                            Material                                  ______________________________________                                        1      -.668   f    .018 f                                                                              1.43500                                                                              95.06                                                                              Calcium Fluoride                        2      -.268   f    .032 f            Air                                     3      -.250   f    .010 f                                                                              1.65426                                                                              58.15                                                                              Hoya LAC7                               4      -.565   f    .001 f            Air                                     5      .455    f    .010 f                                                                              1.65426                                                                              58.15                                                                              Hoya LAC7                               6      .384    f    .001 f            Air                                     7      .424    f    .015 f                                                                              1.43500                                                                              95.06                                                                              Calcium Fluoride                        8      -.635   f                                                              ______________________________________                                    

Two-Glass Mangin Mirrors

FIG. 21 illustrates a particular example of a cemented Mangin doubletmirror, which is color-corrected at five wavelengths. The Mangin doubletmirror of FIG. 21 has unit focal length at λ₀ =0.54607 μm for relativeaperture of f/2, and comprises a first lens element made of Hoya FCD10glass and a second lens element made of Hoya FC3 glass. The outwardfacing surface of the second lens element is provided with a reflectivecoating, which could comprise a conventional reflecting material suchas, e.g., silver or aluminum.

The Hoya FCD10 and Hoya FC3 glasses, used in combination, provide colorcorrection at five wavelengths. The design form of the particular Mangindoublet mirror shown in FIG. 21 is specified in tabular format asfollows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                            Material                                ______________________________________                                        1      -.958   f    .036  f   1.45770                                                                              90.32                                                                              Hoya FCD10                          2      4.200   f    .057  f   1.46619                                                                              65.49                                                                              Hoya FC3                            3      -1.572  f    -.057 f   1.46619                                                                              65.49                                                                              Hoya FC3                            4      4.200   f    -.036 f   1.45770                                                                              90.32                                                                              Hoya FCD10                          5      -.958   f                                                              ______________________________________                                    

where the negative values of thickness listed for the lens element afterthe third surface, as numbered in accordance with conventional practicefrom left to right in the drawing, indicate that reflection occurs atthe third surface. Thus, the fourth surface is the same as the secondsurface, and the fifth surface is the same as the first surface. Lightat a color-corrected wavelength passing consecutively through the firstand second surfaces is reflected from the third surface backconsecutively through the second (i.e., fourth) and first (i.e., fifth)surfaces to the focal plane of the Mangin doublet mirror.

The wavelength scan for the Mangin doublet mirror shown in FIG. 21indicates that precise color correction occurs at five discretewavelengths in the optical spectrum, and that a high degree of axialcolor correction is achieved over a band of wavelengths extending fromthe far ultraviolet to the near infrared regions of the spectrum. Ingeneral, in accordance with the present invention, dispersive effectscan be substantially eliminated from Mangin mirrors by the proper choiceof optical materials from which the lens elements are made.

FIG. 22 illustrates another example of a cemented Mangin doublet mirror,which provides substantially complete axial color correction from theblue to the near infrared regions of the spectrum. The Mangin doubletmirror of FIG. 22 has unit focal length at λ₀ =0.54607 μm for a relativeaperture of f/2, and comprises a first lens element made of Hoya BAF11glass and a second lens element made of Hoya FL6 glass. The outwardfacing surface of the second lens element is provided with a reflectivecoating. The design form of the Mangin doublet mirror shown in FIG. 22is specified in tabular format as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                            Material                                ______________________________________                                        1      -1.897  f    .025  f   1.67000                                                                              47.97                                                                              Hoya BAF11                          2      3.607   f    .040  f   1.57047                                                                              42.57                                                                              Hoya FL6                            3      -1.825  f    -.040 f   1.57047                                                                              42.57                                                                              Hoya FL6                            4      3.607   f    -.025 f   1.67000                                                                              47.97                                                                              Hoya BAF11                          5      -1.897  f                                                              ______________________________________                                    

The wavelength scan for the Mangin doublet mirror of FIG. 22 shows thatprecise axial color correction (i.e., crossing of the horizontal axis)occurs at four discrete wavelengths, and that a high degree of axialcolor correction (i.e., only insignficant paraxial marginal ray height)occurs throughout the spectral range extending from the blue to beyond0.8 μm in the near infrared regions of the spectrum.

FIG. 23 illustrates an example of a cemented Mangin doublet mirror,which provides substantially complete color correction from the blue towell into the near infrared regions of the optical spectrum. The Mangindoublet mirror of FIG. 23 has unit focal length at λ₀ =0.54607 μm forrelative aperture of f/2, and comprises a first lens element made ofCorning-France 486817 glass and a second lens element made ofCorning-France 697554 glass. The outward facing surface of the secondlens element is provided with a reflective coating. The design form ofthe Mangin doublet mirror shown in FIG. 23 is specified in tabularformat as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1      -.753   f    .042  f   1.48746                                                                             81.40                                                                              Corn-F 486817                        2      -7.400  f    .066  f   1.70000                                                                             55.18                                                                              Corn-F 697554                        3      -1.536  f    -.066 f   1.70000                                                                             55.18                                                                              Corn-F 697554                        4      -7.400  f    -.042 f   1.48746                                                                             81.40                                                                              Corn-F 486817                        5      -.753   f                                                              ______________________________________                                    

The wavelength scan for the Mangin doublet mirror of FIG. 23 shows thatprecise axial color correction occurs at four discrete wavelengths, andthat a high degree of axial color correction occurs throughout thespectral range extending from the blue to beyond 1.0 μm in the rearinfrared region of the spectrum.

FIG. 24 illustrates a particular example of an air-spaced Mangin doubletmirror, which is color-corrected at five wavelengths. The Mangin doubletmirror of FIG. 24 has unit focal length at λ₀ =0.54607 μm for a relativeaperture of f/2, and comprises a first lens element made of Schott PK51glass and a second lens element made of Hoya LAC7 glass. The two lenselements are separated by an air gap, and the outward facing surface ofthe second lens element is provided with a reflective coating. Thedesign form of the Mangin doublet mirror shown in FIG. 24 is specifiedin tabular format as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                            Material                                ______________________________________                                        1      -.775   f    .040  f   1.53019                                                                              76.57                                                                              Schott PK51                         2      3.799   f    .007  f               Air                                 3      4.568   f    .061  f   1.65426                                                                              58.15                                                                              Hoya LAC7                           4      -1.501  f    -.061 f   1.65426                                                                              58.15                                                                              Hoya LAC7                           5      4.568   f    -.007 f               Air                                 6      3.799   f    -.040 f   1.53019                                                                              76.57                                                                              Schott PK51                         7      -.775   f                                                              ______________________________________                                    

The wavelength scan for the Mangin doublet mirror of FIG. 24 indicatesthat precise color correction occurs at five discrete wavelengths, andthat a high degree of axial color correction occurs throughout thespectral range extending from the violet to the near infrared regions ofthe optical spectrum.

FIG. 25 illustrates an example of an air-spaced Mangin doublet mirror,which provides substantially complete axial color correction throughoutthe visible and into the near infrared portions of the optical spectrum.The Mangin doublet mirror of FIG. 25 has unit focal length at λ₀=0.54607 μm for a relative aperture of f/2, and comprises a first lenselement made of Hoya FCD10 glass and a second lens element made of OharaPSK1 glass. The two lens elements are separated by an air gap, and theoutward facing surface of the second lens element is provided with areflective coating. The design form of the Mangin doublet mirror shownin FIG. 25 is specified in tabular format as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                            Material                                ______________________________________                                        1      -.736   f    .036  f   1.45770                                                                              90.32                                                                              Hoya FCD10                          2      8.012   f    .004  f               Air                                 3      8.273   f    .057  f   1.54979                                                                              62.60                                                                              Ohara PSK1                          4      -1.476  f    -.057 f   1.54979                                                                              62.60                                                                              Ohara PSK1                          5      8.273   f    -.004 f               Air                                 6      8.012   f    -.036 f   1.45770                                                                              90.32                                                                              Hoya FCD10                          7      -.736   f                                                              ______________________________________                                    

The wavelength scan curve shown in FIG. 25 substantially coincides withthe horizontal axis throughout the visible portion of the opticalspectrum, although a larger scale plot would show that the crossings ofthe horizontal axis actually occur at five discrete wavelengths.Technically, therefore, the Mangin doublet mirror of FIG. 25 iscolor-corrected at only five wavelengths. However, for many practicalapplications, this Mangin doublet mirror can be considered to besubstantially color-corrected throughout the visible wavelength band.

FIG. 26 illustrates an example of an air-spaced Mangin doublet mirror,which provides substantially complete axial color correction from thegreen region of the visible spectrum to well into the infraredwavelengths. The Mangin doublet mirror of FIG. 26 has unit focal lengthat λ₀ =0.54607 μm for a relative aperture of f/2, and comprises a firstlens element made of Hoya FCD10 glass and a second lens element made ofHoya LAC14 glass. The two lens elements are separated by an air gap, andthe outward facing surface of the second lens element is provided with areflective coating. The design form of the Mangin doublet mirror shownin FIG. 26 is specified in tabular format as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                            Material                                ______________________________________                                        1      -.661   f    .029  f   1.45770                                                                              90.32                                                                              Hoya FCD10                          2      -2.254  f    .010  f               Air                                 3      -3.628  f    .054  f   1.69980                                                                              55.25                                                                              Hoya LAC14                          4      -1.519  f    -.054 f   1.69980                                                                              55.25                                                                              Hoya LAC14                          5      -3.628  f    -.010 f               Air                                 6      -2.254  f    -.029 f   1.45770                                                                              9.32 Hoya FCD10                          7      -.661   f                                                              ______________________________________                                    

The wavelength scan curve shown in FIG. 26 substantially coincides withthe horizontal axis from the green wavelength region to beyond 1.0 μm inthe near infrared wavelength region of the spectrum. A larger scale plotwould show that crossings of the horizontal axis actually occur at onlyfour discrete wavelengths. Technically, therefore, the Mangin doubletmirror of FIG. 26 is color-corrected at only four wavelengths. However,for many practical applications, particularly applications in the nearinfrared region of the spectrum, the Mangin doublet mirror of FIG. 26can be considered to be substantially color-corrected throughout thewavelength band of interest.

Two-Glass Petzval Lens Systems

The designation "Petzval lens system" is generally applied to a lenssystem comprising two widely separated lens groups, each of which has anet positive dioptric power, followed by a negative lens element or alens group having a net negative dioptric power. The positive lensgroups in combination provide good resolution at high aperture fornarrow to moderate fields of view. The negative lens element or lensgroup functions as a field flattener. A lens group is said to havepositive dioptric power when it causes light rays passing through it toconverge, and to have negative dioptric power when it causes light rayspassing through it to diverge. Often, each lens group of a Petzval lenssystems is a doublet. However, for a particular application, either one(or all) of the lens groups of a Petzval lens system could be a singlet,a triplet or a high-number multiplet.

The spacing between the two positive lens groups of a Petzval lenssystem is large in comparison with the thicknesses of the individuallens elements comprising each lens group. Within a lens group, theindividual lens elements could be cemented or air-spaced. It is usualfor a negative lens element or lens group functioning as a fieldflattener to be positioned immediately in front of the image plane ofthe system to correct for field curvature. A Petzval lens system thatincludes a field flattener is called, appropriately, a flat-fieldPetzval lens system.

For a given optical design, aberrations scale linearly with the physicalsize of the system. Thus, a lens system that is reasonablywell-corrected for a relatively short focal length (say, less than 10mm) might have very severe aberration problems at a focal length of morethan 100 mm. In the prior art, Petzval lens systems were typically usedonly in short-focus applications, e.g., in projectors for 16 mm and 8 mmmotion-picture films. For very long-focus applications (e.g., more than200 mm), Petzval-type lens systems were not generally used because ofthe difficulty in designing high-aperture long-focus Petzval lenssystems that are well-corrected.

The lens groups of Petzval lens systems of the prior art were typically"achromats", i.e., doublets color-corrected at only two wavelengths.Furthermore, Petzval lens systems of the prior art were typically usedonly in relatively narrow field-angle applications. An example of ashort-focus narrow field-angle Petzval lens system is described in U.S.Pat. No. 3,255,664.

FIG. 27 shows a two-glass flat-field lens system of the Petzval typemade of the same two types of optical glasses as used for the lensdoublet shown in FIG. 12, i.e., Schott PK51 and Hoya LAC7 glasses. Thelast lens element in the system, which is a biconcave lens made of HoyaLAC7 glass, functions as a field flattener. The Petzval lens system ofFIG. 27 is color-corrected at five wavelengths in the optical spectrumfrom 0.4 μm to 0.7 μm, and is also well-corrected for monochromaticaberrations in this spectral range.

The design form of the Petzval lens system of FIG. 27, in terms of thefocal length for a relative aperture of f/3.5, is specified in tabularformat as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                            Material                                ______________________________________                                        1      .342    f    .031 f  1.53019                                                                              76.57                                                                              Schott PK51                           2      -.497   f    .006 f              Air                                   3      -1.190  f    .020 f  1.65426                                                                              58.15                                                                              Hoya LAC7                             4      1.465   f    .015 f              Air                                   5      -.299   f    .020 f  1.65426                                                                              58.15                                                                              Hoya LAC7                             6      .522    f    .001 f              Air                                   7      .350    f    .028 f  1.53019                                                                              76.57                                                                              Schott PK51                           8      -.438   f    .274 f              Air                                   9      .508    f    .023 f  1.53019                                                                              76.57                                                                              Schott PK51                           10     -.593   f    .016 f              Air                                   11     -.225   f    .018 f  1.65426                                                                              58.15                                                                              Hoya LAC7                             12     1.483   f    .014 f              Air                                   13     .417    f    .025 f  1.53019                                                                              76.57                                                                              Schott PK51                           14     -.356   f    .268 f              Air                                   15     -.241   f    .013 f  1.65426                                                                              58.15                                                                              Hoya LAC7                             16     2.412   f                                                              ______________________________________                                    

The wavelength scan for the Petzval lens system of FIG. 27 shows threecrossings of the horizontal axis in the visible spectral range, as wellas single crossings in the ultraviolet and infrared portions of thespectrum. For the visible wavelengths from the blue region into the redregion, the paraxial marginal ray height is insignificant, indicatingthat very substantial axial color correction occurs throughout thatportion of the visible spectrum. The design form of this Petzval lenssystem also provides superb correction of the monochromatic aberrations,and the chromatic variation of spherical aberration and coma.

The Petzval lens system of FIG. 27 is noteworthy for having a long focallength (f=600 mm) at a relative aperture of f/3.5, while providing afield of view of 12°. This particular Petzval lens system is withoutprecedent in enabling diffraction limited performance to be provided atsuch a large aperture and wide field of view.

FIG. 28 shows a two-glass flat-field lens system of the Petzval typemade of the same two types of optical glasses as used for the lensdoublets shown in FIGS. 10 and 11, i.e., Corning-France 486817 and OharaLAK14 glasses. The Petzval lens system of FIG. 28 is color-corrected atfour wavelengths, two in the visible region and two in the infraredregion. This system is also well-corrected for spherochromatism and themonochromatic aberrations.

The design form of the Petzval lens system of FIG. 28, in terms of thefocal length for a relative aperture of f/3.5, is specified in tabularformat as follows:

    ______________________________________                                        Surface                                                                       No.    Radius   Thickness N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1      .305    f    .020 f  1.48746                                                                             81.40                                                                              Corn-F 486817                          2      -.900   f    .005 f             Air                                    3      -.337   f    .013 f  1.69979                                                                             55.31                                                                              Ohara LAK14                            4      .398    f    .001 f             Air                                    5      .192    f    .018 f  1.48746                                                                             81.40                                                                              Corn-F 486817                          6      -.692   f    .166 f             Air                                    7      .441    f    .016 f  1.48746                                                                             81.40                                                                              Corn-F 486817                          8      -.385   f    .010 f             Air                                    9      -.146   f    .011 f  1.69979                                                                             55.31                                                                              Ohara LAK14                            10     -.552   f    .009 f             Air                                    11     .393    f    .060 f  1.48746                                                                             81.40                                                                              Corn-F 486817                          12     -.240   f    .187 f             Air                                    13     -.162   f    .014 f  1.69979                                                                             55.31                                                                              Ohara LAK14                            14     .789    f                                                              ______________________________________                                    

The wavelength scan for the Petzval lens system of FIG. 28 shows verysubstantial (if not perfect) color-correction throughout the spectralrange extending from the blue wavelength region to beyond 1.0 μm in thenear infrared region. It is remarkable that such color correction can beobtained for a Petzval lens system using only two different opticalglasses. The Petzval lens system of FIG. 28 has a long focal length(f=380 mm) at a relative aperture of f/3.5, while providing a field ofview of approximately 12°. The design form of this Petzval lens systemprovides diffraction limited performance throughout the spectral rangefor which it is color-corrected.

FIG. 29 shows another long-focus flat-field Petzval lens system whoselens elements are made of only two different types of optical glasses,viz., Schott FK51 and Schott LAKN14 glasses. This Petzval lens system iscolor-corrected at three wavelengths, two in the visible region and onein the infrared region. The system is also well-corrected forspherochromatism and the monochromatic aberrations, and providesdiffraction limited performance throughout the spectral range for whichit is color-corrected.

The design form of the Petzval lens system of FIG. 29, in terms of thefocal length for a relative aperture of f/2 (i.e., an extremely wideaperture), is specified in tabular format as follows:

    ______________________________________                                        Surface          Thick-                                                       No.    Radius    ness    N.sub.e                                                                             V.sub.e                                                                            Material                                  ______________________________________                                         1     2.066 f   .094 f  1.48794                                                                             84.07                                                                              Schott FK51                                2     -1.933 f  .156 f             Air                                        3     4.568 f   .087 f  1.48794                                                                             84.07                                                                              Schott FK51                                4     -1.026 f  .004 f             Air                                        5     -.991 f   .037 f  1.69980                                                                             55.19                                                                              Schott LAKN14                              6     24.483 f  .033 f             Air                                        7     -1.339 f  .037 f  1.69980                                                                             55.19                                                                              Schott LAKN14                              8     63.427 f  .001 f             Air                                        9     .850 f    .087 f  1.48794                                                                             84.07                                                                              Schott FK51                               10     -5.545 f  .873 f             Air                                       11     .422 f    .050 f  1.48794                                                                             84.07                                                                              Schott FK51                               12     -.555 f   .010 f             Air                                       13     -.473 f   .025 f  1.69980                                                                             55.19                                                                              Schott LAKN14                             14     -5.724 f  .032 f             Air                                       15     -2.021 f  .025 f  1.69980                                                                             55.19                                                                              Schott LAKN14                             16     .969 f    .001 f             Air                                       17     .340 f    .050 f  1.48794                                                                             84.07                                                                              Schott FK51                               18     -.798 f   .249 f             Air                                       19     -.252 f   .025 f  1.69980    Schott LAKN14                             20     5.189 f                                                                ______________________________________                                    

The wavelength scan for the Petzval lens system of FIG. 29 shows thatprecise axial color-correction occurs at the wavelengths 0.527 μm, 0.640μm, and 0.778 μm. In the range from 0.527 μm to 0.778 μm, the degree ofcolor-correction obtainable is very high. The Petzval lens system ofFIG. 29 has a focal length of 1000 mm, and a field angle of 4.2°.

Two-Glass Prism Systems

FIG. 30 illustrates a two-glass cemented doublet prism system having adeviation angle of 6° for which very substantial color correction (i.e.,only insignificant dispersion) occurs throughout the visible spectrum.The deviation angle of a prism is defined as the angle through which aray changes direction after passing through the prism, i.e., the anglebetween the direction of the ray entering the system and the directionof the ray exiting from the system. The prism system of FIG. 30comprises a first prism element made of Schott PK51 glass and having anapex angle of α₁ =42.0239°, and a second prism element made of Hoya LAC7glass and having an apex angle of α₂ =-25.5872°. The negative sign forα₂ is a convention indicating that the orientation of the apex of thesecond prism element is inverted with respect to the orientation of theapex of the first prism element.

Also shown in FIG. 30 is a plot of change in angular deviation (indegrees) versus wavelength (in micrometers) for this particular doubletprism system for a 6° total deviation angle. The curve in FIG. 30crosses the horizontal axis at five discrete wavelengths, whichindicates that chromatic dispersion from a total deviation angle of 6°is precisely zero at those discrete wavelengths. It is particularlynoteworthy, however, that the curve in FIG. 30 is very close to thehorizontal axis throughout the entire visible region extending well intothe infrared region of the spectrum to about 0.8 μm. For thosewavelengths at which the total deviation angle differs from 6°, thedifference is approximately 0.0001 degree throughout this wavelengthband. Thus, for many applications, the prism system of FIG. 30 can beconsidered to be color-corrected (i.e., as having substantially zerodispersion) for a deviation angle of 6° throughout this rather widewavelength band, which includes the entire visible region.

The plot of change in angular deviation versus wavelength shown in FIG.30 describes a "real" ray trace, not a paraxial ray trace as in theforegoing plots for dioptric and catadioptric systems. The plots shownin FIG. 30 and in the following examples of other prism systems werebased upon data obtained from ray traces according to Snell's law.

FIG. 31 illustrates a two-glass cemented quintuplet prism system, whichprovides a high degree of color correction throughout the visiblespectrum for a total deviation angle of 30°. The prism system of FIG. 31comprises a first prism element made of Schott PK51 glass and having anapex angle of α₁ =43.1695°, a second prism element made of Hoya LAC7glass and having an apex angle of α₂ =-43.1695°, a third prism elementmade of Schott PK51 glass and having an apex angle of α₃ =43.1695°, afourth prism element made of Hoya LAC7 glass and having an apex angle ofα₄ =-43.1695°, and a fifth prism element made of Schott PK51 glass andhaving an apex angle of α₅ =57.5776°. From a manufacturing standpoint,it is of special interest that α₁, α₂, α₃ and α₄ having the same value,where the negative sign given for α₂ and α₄ indicates that theorientations of the second and fourth prism elements are inverted withrespect to the orientations of the first, third and fifth prismelements.

A plot of change in angular deviation versus wavelength shown in FIG. 31indicates that chromatic dispersion from a total deviation angle of 30°is precisely zero at five discrete wavelengths. It is to be noted that atotal deviation angle of 30° is a large total deviation angle. The 30°deviation of a beam of light that is produced by the prism system ofFIG. 31 is equivalent to the deviation produced by the marginal zone ofa lens with a relative aperture of f/1.0. The plot of change in angulardeviation versus wavelength in FIG. 31 indicates that the change intotal angular deviation from 30° is only about 0.0005 degree for anywavelength in the spectral band that includes the entire visible regionand a portion of the near infrared region extending to almost 0.8 μm.The change in total angular deviation from 30° is in fact zero at fivediscrete wavelengths in this spectral band.

FIG. 32 illustrates a two-glass cemented quintuplet prism system, whichis designed using the same types of glasses as the quintuplet prismsystem of FIG. 31. The only difference between the prism system of FIG.32 and the prism system of FIG. 31 is that the apex angles of the prismelements of the system shown in FIG. 32 have different values. Thus, forthe first prism element α₁ =83.1964°, for the second prism element α₂=-44.9553°, for the third prism element α₃ =40.8576°, for the fourthprism element α₄ =-46.9210°, and for the fifth prism element α₅=37.5019°.

Optical Systems Comprising More Than Two Glasses

The present invention provides the designer with the ability to designoptical systems that are color-corrected at three or more wavelengthsusing only two different optical materials. Thus, as a result of thepresent invention, complicated multiglass optical systems of the priorart can be replaced by simpler two-glass optical systems that providecomparable, or better, color correction. This ability to simplifycomplicated optical systems while improving performance is a majoradvantage inherent in the present invention.

The method of the present invention for selecting more than two opticalmaterials for designing color-corrected optical systems is of generalinterest in terms of demonstrating the unified theory. Ordinarily, inpractical applications, a two-glass optical system is to be preferred toa system having the same capabilities but comprising more than twooptical materials. In particular applications, however, where colorcorrection at particular wavelengths is required, optical systemscomprising three or more different optical materials might be used toprovide optimum color correction for specified operational requirements.

FIG. 33 illustrates a particular example of a three-glass lens multipletfor which the lens elements comprising the multiplet were selected inaccordance with the present invention. The lens multiplet of FIG. 33 isa quintuplet having a focal length of 1000 mm at λ₀ =0.54607 μm for arelative aperture of f/10, and provides color correction at fivewavelengths. In accordance with conventional practice, the five lenselements comprising the lens quintuplet are described consecutively fromleft to right as shown in drawing. The design form of the lensquintuplet of FIG. 33 is specified in tabular format as follows:

    ______________________________________                                        Surface          Thick-                                                       No.    Radius    ness     N.sub.e                                                                             V.sub.e                                                                            Material                                 ______________________________________                                        1      .493 f    .015 f   1.49845                                                                             81.22                                                                              Ohara FK01                               2      -.660 f   .034 f              Air                                      3      -.538 f   .012 f   1.72538                                                                             34.49                                                                              Ohara KZFS8                              4      1.078 f   .012 f   1.49845                                                                             81.22                                                                              Ohara FK01                               5      -2.457 f  .001 f              Air                                      6      -5.045 f  .013 f   1.70443                                                                             29.89                                                                              Ohara SF15                               7      -.288 f   .013 f   1.72538                                                                             34.49                                                                              Ohara KZFS8                              8      -.953 f                                                                ______________________________________                                    

The wavelength scan for the three-glass multiplet shown in FIG. 33indicates that a lens system can be designed using Ohara FK01, OharaKZFS8 and Ohara FK01 glasses for bringing five wavelength from theultraviolet through the visible to the near infrared regions of thespectrum to a common focus. The wavelength scan further indicates thatsubstantially complete axial color correction can be provided by thesethree glasses in combination for wavelengths in the near infrared regionup to at least 1.0 μm.

FIG. 34 illustrates a particular example of a four-glass lens multipletfor which the lens elements comprising the multiplet were selected inaccordance with the present invention. The lens multiplet of FIG. 34 isa quintuplet having a focal length of 1000 mm at λ₀ =0.54607 μm for arelative aperture of f/10, and provides color correction at fivewavelengths. The design form of the lens quintuplet of FIG. 34 isspecified in tabular format as follows:

    ______________________________________                                        Surface          Thick-                                                       No.    Radius    ness    N.sub.e                                                                             V.sub.e                                                                            Material                                  ______________________________________                                        1      .425 f    .015 f  1.53019                                                                             76.57                                                                              Schott PK51                               2      -.749 f   .008 f             Air                                       3      -.369 f   .012 f  1.69669                                                                             51.28                                                                              Schott LAK20                              4      .416 f    .015 f  1.53019                                                                             76.57                                                                              Schott PK51                               5      -.385 f   .001 f             Air                                       6      .546 f    .015 f  1.61669                                                                             44.07                                                                              Schott KZFSN4                             7      -.170 f   .012 f  1.59487                                                                             48.29                                                                              Schott KZFS6                              8      .434 f                                                                 ______________________________________                                    

The wavelength scan for the four-glass multiplet shown in FIG. 34indicates that a lens system can be designed using Schott PK51, SchottLAK20, Schott KZFSN4 and Schott KZFS6 glasses for bringing fivewavelengths from the ultraviolet through the visible to the nearinfrared regions of the spectrum to a common focus. The wavelength scanfurther indicates that substantially complete color correction can beprovided by these four glasses in combination for wavelengths in therange from about 0.60 μm to 0.85 μm.

CONCLUSION

This invention has been described above in terms of particular opticalsystems designed using particular glasses and other types of opticalmaterials that are commercially available at the present time. However,other kinds of optical systems, which could be designed using othertypes of glasses and optical materials, could be made by designersskilled in the art on the basis of the foregoing disclosure inconjunction with the accompanying drawing. Therefore, the novel opticalsystems described above are to be considered as merely illustrative ofthe invention. The legal definition of the invention is provided moregenerally by the following claims and their equivalents.

                                      TABLE I                                     __________________________________________________________________________    LEAST SQUARES FITTED COEFFICIENTS                                             ORDER      BASE WVL         MIN VL  MAX WVL                                   2.0        0.5740           0.400   0.700                                     CAT   NAME ETA1/ETA2                                                                              INDEX   ETA1    ETA2    ARCTAN(ETA1/ETA2)                 __________________________________________________________________________    OHARA SK4  -.500727 + 05                                                                          1.613471                                                                              -.939625 - 01                                                                         .187652 - 05                                                                          -89.998856                        OHARA LAK010                                                                             -.324490 + 04                                                                          1.726980                                                                              -.102990 + 00                                                                         .317391 - 04                                                                          -89.982343                        OHARA LAK11                                                                              -.294092 + 04                                                                          1.659127                                                                              -.962812 - 01                                                                         .327384 - 04                                                                          -89.980517                        CHANCE                                                                              691547                                                                             -.192659 + 04                                                                          1.691914                                                                              -.100937 + 00                                                                         .523916 - 04                                                                          -89.970261                        SCHOTT                                                                              SK15 -.148992 + 04                                                                          1.623767                                                                              -.951161 - 01                                                                         .638397 - 04                                                                          -89.961544                        OHARA ZK1  -.113189 + 04                                                                          1.533817                                                                              -.952528 - 01                                                                         .841541 - 04                                                                          -89.949380                        SCHOTT                                                                              SK51 -.882619 + 03                                                                          1.621646                                                                              -.915519 - 01                                                                         .103728 - 03                                                                          -89.935084                        SCHOTT                                                                              BAK5 -.863211 + 03                                                                          1.557397                                                                              -.941415 - 01                                                                         .109060 - 03                                                                          -89.933624                        SCHOTT                                                                              ZK1  -.542770 + 03                                                                          1.533815                                                                              -.952493 - 01                                                                         .175487 - 03                                                                          -89.894439                        OHARA LAK8 -.505624 + 03                                                                          1.713954                                                                              -.102540 + 00                                                                         .202799 - 03                                                                          -89.886683                        CORN-F                                                                              713538                                                                             -.420221 + 03                                                                          1.713958                                                                              -.102594 + 00                                                                         .244142 - 03                                                                          -89.863654                        HOYA  LAC12                                                                              -.391017 + 03                                                                          1.678786                                                                              -.994099 - 01                                                                         .254234 - 03                                                                          -89.853470                        OHARA SK15 -.360639 + 03                                                                          1.623767                                                                              -.949577 - 01                                                                         .263304 - 03                                                                          -89.841127                        HOYA  C12  -.346402 + 03                                                                          1.523719                                                                              -.941556 - 01                                                                         .271810 - 03                                                                          -89.834599                        HOYA  C3   -.307862 + 03                                                                          1.518870                                                                              -.936381 - 01                                                                         .304156 - 03                                                                          -89.813891                        OHARA ZK4  -.297025 + 03                                                                          1.512537                                                                              -.949230 - 01                                                                         .319579 - 03                                                                          -89.807101                        SCHOTT                                                                              SK13 -.245242 + 03                                                                          1.592544                                                                              -.946779 - 01                                                                         .386059 - 03                                                                          -89.766372                        CHANCE                                                                              572577                                                                             -.238564 + 03                                                                          1.572917                                                                              -.956311 - 01                                                                         .400862 - 03                                                                          -89.759831                        CORN-F                                                                              651559                                                                             -.207410 + 03                                                                          1.651968                                                                              -.987661 - 01                                                                         .476189 - 03                                                                          -89.723757                        OHARA BAK1 -.203696 + 03                                                                          1.573218                                                                              -.956013 - 01                                                                         .469332 - 03                                                                          -89.718722                        HOYA  BACD13                                                                             -.195377 + 03                                                                          1.592545                                                                              -.946756 - 01                                                                         .484579 - 03                                                                          -89.706745                        CORN-F                                                                              658572                                                                             -.193118 + 03                                                                          1.658335                                                                              -.965192 - 01                                                                         .499794 - 03                                                                          -89.703314                        OHARA K2   -.172343 + 03                                                                          1.516673                                                                              -.972221 - 01                                                                         .564120 - 03                                                                          -89.667551                        OHARA BAK5 -.165965 + 03                                                                          1.557399                                                                              -.941142 - 01                                                                         .567071 - 03                                                                          -89.654778                        OHARA KO1  -.146523 + 03                                                                          1.523717                                                                              -.944011 - 01                                                                         .644277 - 03                                                                          -89.608969                        OHARA LAK12                                                                              -.138588 + 03                                                                          1.678787                                                                              -.997704 - 01                                                                         .719909 - 03                                                                          -89.586580                        CORN-F                                                                              691548                                                                             -.127310 + 03                                                                          1.691921                                                                              -.100767 + 00                                                                         .791505 - 03                                                                          -89.549962                        SCHOTT                                                                              SK52 -.953341 + 02                                                                          1.639373                                                                              -.993953 - 01                                                                         .104260 - 02                                                                          -89.399022                        OHARA SK10 -.921555 + 02                                                                          1.623589                                                                              -.966943 - 01                                                                         .104925 - 02                                                                          -89.378296                        OHARA LAK13                                                                              -.907372 + 02                                                                          1.694441                                                                              -.103439 + 00                                                                         .113999 - 02                                                                          -89.368575                        OHARA LAKO11                                                                             -.897967 + 02                                                                          1.742014                                                                              -.104780 + 00                                                                         .116686 - 02                                                                          -89.361963                        OHARA LAKO13                                                                             -.830518 + 02                                                                          1.641813                                                                              -.969394 - 01                                                                         .116722 - 02                                                                          -89.310151                        SCHOTT                                                                              K10  -.824042 + 02                                                                          1.502014                                                                              -.978985 - 01                                                                         .118803 - 02                                                                          -89.304731                        OHARA SK19 -.787332 + 02                                                                          1.614193                                                                              -.961291 - 01                                                                         .122095 - 02                                                                          -89.272317                        SCHOTT                                                                              SK19 -.783834 + 02                                                                          1.614195                                                                              -.961830 - 01                                                                         .122708 - 02                                                                          -89.269073                        HOYA  BACD10                                                                             -.768258 + 02                                                                          1.623591                                                                              -.969209 - 01                                                                         .126157 - 02                                                                          -89.254251                        SCHOTT                                                                              BAK1 -.751939 + 02                                                                          1.573219                                                                              -.958644 - 01                                                                         .127490 - 02                                                                          -89.238069                        HOYA  BACD1                                                                              -.743045 + 02                                                                          1.611028                                                                              -.974048 - 01                                                                         .131089 - 02                                                                          -89.228950                        HOYA  LACL2                                                                              -.727112 + 02                                                                          1.651040                                                                              -.992233 - 01                                                                         .136462 - 02                                                                          -89.212060                        HOYA  BACD6                                                                              -.716787 + 02                                                                          1.614541                                                                              -.978361 -  01                                                                        .136493 - 02                                                                          -89.200707                        SCHOTT                                                                              LAKN22                                                                             -.715624 + 02                                                                          1.651973                                                                              -.987132 - 01                                                                         .137940 - 02                                                                          -89.199411                        CHANCE                                                                              623569                                                                             -.657657 + 02                                                                          1.623589                                                                              -.969717 - 01                                                                         .147450 - 02                                                                          -89.128858                        SCHOTT                                                                              SK10 -.654909 + 02                                                                          1.623593                                                                              -.969513 - 01                                                                         .148038 - 02                                                                          -89.125200                        HOYA  SBF5 -.591198 + 02                                                                          1.522013                                                                              -.104595 + 00                                                                         .176920 - 02                                                                          -89.030948                        OHARA LASKO1                                                                             -.581632 + 02                                                                          1.756042                                                                              -.105438 + 00                                                                         .181279 - 02                                                                          -89.015014                        SCHOTT                                                                              SK2  -.566100 + 02                                                                          1.608157                                                                              -.973525 - 01                                                                         .171971 - 02                                                                          -88.987988                        CORN-F                                                                              623569                                                                             -.561077 + 02                                                                          1.623593                                                                              -.969182 - 01                                                                         .172736 - 02                                                                          -88.978932                        CHANCE                                                                              569561                                                                             -.560878 + 02                                                                          1.569560                                                                              -.982912 - 01                                                                         .175245 - 02                                                                          -88.978572                        CHANCE                                                                              678552                                                                             -.556141 + 02                                                                          1.678788                                                                              -.999158 - 01                                                                         .179659 - 02                                                                          -88.969873                        SCHOTT                                                                              SK1  -.524111 + 02                                                                          1.611029                                                                              -.972537 - 01                                                                         .185559 - 02                                                                          -88.906936                        CORN-F                                                                              614564                                                                             -.520142 + 02                                                                          1.614285                                                                              -.977645 - 01                                                                         .187957 - 02                                                                          -88.898597                        HOYA  BACD2                                                                              -.519914 + 02                                                                          1.608159                                                                              -.972569 - 01                                                                         .187064 - 02                                                                          -88.898108                        HOYA  ADC1 -.515199 + 02                                                                          1.620722                                                                              -.886488 - 01                                                                         .172067 - 02                                                                          -88.888030                        HOYA  BAC4 -.500459 + 02                                                                          1.569563                                                                              -.984724 - 01                                                                         .196764 - 02                                                                          -88.855289                        OHARA BAK6 -.497699 + 02                                                                          1.575177                                                                              -.976194 - 01                                                                         .196141 - 02                                                                          -88.848944                        SCHOTT                                                                              LAKN16                                                                             -.481711 + 02                                                                          1.734527                                                                              -.106819 + 00                                                                         .221749 - 02                                                                          -88.810749                        CHANCE                                                                              691548                                                                             -.474013 + 02                                                                          1.691911                                                                              -.100650 + 00                                                                         .212337 - 02                                                                          -88.791434                        SCHOTT                                                                              SK6  -.472162 + 02                                                                          1.614537                                                                              -.977687 - 01                                                                         .207066 - 02                                                                          -88.786704                        OHARA SK1  -.470107 + 02                                                                          1.611035                                                                              -.975581 - 01                                                                         .207523 - 02                                                                          -88.781403                        SCHOTT                                                                              BAK4 -.466145 + 02                                                                          1.569563                                                                              -.982860 - 01                                                                         .210849 - 02                                                                          -88.771045                        OHARA BAK4 -.465233 + 02                                                                          1.569563                                                                              -.982470 -  01                                                                        .211178 - 02                                                                          -88.768640                        OHARA LAKO4                                                                              -.461629 + 02                                                                          1.651835                                                                              -.982167 - 01                                                                         .212761 - 02                                                                          -88.759029                        SCHOTT                                                                              BAK6 -.446430 + 02                                                                          1.575176                                                                              -.978150 - 01                                                                         .219105 - 02                                                                          -88.716792                        OHARA SK6  -.445799 + 02                                                                          1.614537                                                                              -.978164 - 01                                                                         .219418 - 02                                                                          -88.714978                        SCHOTT                                                                              LAKN12                                                                             -.438037 + 02                                                                          1.678788                                                                              -.998759 - 01                                                                         .228008 - 02                                                                          -88.692214                        HOYA  BACD8                                                                              -.438015 + 02                                                                          1.611961                                                                              -.988673 - 01                                                                         .225717 - 02                                                                          -88.692147                        HOYA  BAC1 -.418595 + 02                                                                          1.573217                                                                              -.959622 - 01                                                                         .229248 - 02                                                                          -88.631498                        OHARA SK2  -.415061 + 02                                                                          1.608152                                                                              -.970336 - 01                                                                         .233782 - 02                                                                          -88.619845                        CHANCE                                                                              734517                                                                             -.399053 + 02                                                                          1.734527                                                                              -.106738 + 00                                                                         .267478 - 02                                                                          -88.564508                        SCHOTT                                                                              SK8  -.396332 + 02                                                                          1.611961                                                                              -.985867 - 01                                                                         .248748 - 02                                                                          -88.554654                        OHARA KZF5 -.386798 + 02                                                                          1.522016                                                                              -.104984 + 00                                                                         .271418 - 02                                                                          -88.519047                        OHARA LAKO9                                                                              - .386447 + 02                                                                         1.735028                                                                              -.107216 + 00                                                                         .277441 - 02                                                                          -88.517697                        OHARA KF3  -.382488 + 02                                                                          1.515220                                                                              -.100854 + 00                                                                         .263679 - 02                                                                          -88.502365                        CORN-F                                                                              569560                                                                             -.377078 + 02                                                                          1.569531                                                                              -.984073 - 01                                                                         .260973 - 02                                                                          -88.480890                        CORN-F                                                                              734514                                                                             -.374738 + 02                                                                          1.734545                                                                              -.107304 + 00                                                                         .286344 - 02                                                                          -.88.471407                       OHARA SK8  -.370859 + 02                                                                          1.611956                                                                              -.985702 - 01                                                                         .265789 - 02                                                                          -88.455426                        OHARA ZK5  -.369012 + 02                                                                          1.534444                                                                              -.992690 - 01                                                                         .269013 - 02                                                                          -88.447699                        SCHOTT                                                                              SKN18                                                                              -.361546 + 02                                                                          1.639374                                                                              -.994923 - 01                                                                         .275186 - 02                                                                          -88.415658                        HOYA  TAC4 -.359864 + 02                                                                          1.735038                                                                              -.108040 + 00                                                                         .300226 - 02                                                                          -88.408251                        HOYA  BAC6 -.349833 + 02                                                                          1.575178                                                                              -.978458 - 01                                                                         .279693 - 02                                                                          -88.362641                        HOYA  BACD18                                                                             -.343625 + 02                                                                          1.639374                                                                              -.994040 - 01                                                                         .289281 - 02                                                                          -88.333075                        SCHOTT                                                                              LAK28                                                                              -.331265 + 02                                                                          1.745347                                                                              -.108591 + 00                                                                         .327807 - 02                                                                          -88.270920                        CORN-F                                                                              639555                                                                             -.330270 + 02                                                                          1.639336                                                                              -.993028 - 01                                                                         .300671 - 02                                                                          -88.265717                        HOYA  BACED4                                                                             -.326817 + 02                                                                          1.618461                                                                              -.999157 - 01                                                                         .305724 - 02                                                                          -88.247398                        CORN-F                                                                              529518                                                                             -.323428 + 02                                                                          1.529735                                                                              -.106441 + 00                                                                         .329102 - 02                                                                          -88.229052                        HOYA  TAF1 -.321194 + 02                                                                          1.773625                                                                              -.111086 + 00                                                                         .345853 - 02                                                                          -88.216740                        SCHOTT                                                                              KF3  -.321153 + 02                                                                          1.515220                                                                              -.100799 + 00                                                                         .313867 - 02                                                                          -88.216505                        HOYA  ZNC5 -.317057 + 02                                                                          1.534442                                                                              -.995644 - 01                                                                         .314026 - 02                                                                          -88.193491                        SCHOTT                                                                              KZF2 -.308127 + 02                                                                          1.530180                                                                              -.106733 + 00                                                                         .346394 - 02                                                                          -88.141160                        CHANCE                                                                              639554                                                                             -.303756 + 02                                                                          1.639373                                                                              -.994654 - 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01                                                                          2.577  -10.498                    OHARA SF8  -4.050                                                                            1.688770                                                                           -.161032 + 00                                                                         .397613 - 01                                                                          .175491 - 01                                                                          2.266  -9.176                     SCHOTT                                                                              SF8  -4.045                                                                            1.688772                                                                           -.160524 + 00                                                                         .396840 - 01                                                                          .180977 - 01                                                                          2.193  -8.870                     HOYA  FD8  -4.030                                                                            1.688771                                                                           -.160675 + 00                                                                         .398746 - 01                                                                          .181835 - 01                                                                          2.193  -8.836                     CHANCE                                                                              706300                                                                             -3.904                                                                            1.705688                                                                           -.166420 + 00                                                                         .426291 - 01                                                                          -.125649 - 02                                                                         -33.927                                                                              132.448                    SCHOTT                                                                              SF52 -3.813                                                                            1.688358                                                                           -.163202 + 00                                                                         .428023 - 01                                                                          .179492 - 01                                                                          2.385  -9.092                     CORN-F                                                                              836302                                                                             -3.799                                                                            1.836206                                                                           -.165259 + 00                                                                         .435053 - 01                                                                          .274394 - 02                                                                          15.855 -60.227                    CORN-F                                                                              699301                                                                             -3.746                                                                            1.698834                                                                           -.165739 + 00                                                                         .442431 - 01                                                                          .127899 - 01                                                                          3.459  -12.959                    HOYA  FD15 -3.705                                                                            1.698777                                                                           -.166066 + 00                                                                         .448226 - 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01                                                                          .116099 - 01                                                                          3.864  -14.262                    CHANCE                                                                              699301                                                                             -3.651                                                                            1.698778                                                                           -.166161 + 00                                                                         .455060 - 01                                                                          .144051 - 01                                                                          3.159  -11.535                    OHARA SF1  -3.631                                                                            1.717187                                                                           -.169088 + 00                                                                         .465651 - 01                                                                          .195078 - 01                                                                          2.387  -8.668                     SCHOTT                                                                              SF1  -3.624                                                                            1.717185                                                                           -.169134 + 00                                                                         .466728 - 01                                                                          .194224 -  01                                                                         2.403  -8.708                     HOYA  FD1  -3.597                                                                            1.717184                                                                           -.169250 + 00                                                                         .470500 - 01                                                                          .198544 - 01                                                                          2.370  -8.525                     HOYA  FD20 -3.587                                                                            1.720047                                                                           -.170467 + 00                                                                         .475280 - 01                                                                          .161718 - 01                                                                          2.939  -10.541                    CHANCE                                                                              717295                                                                             -3.578                                                                            1.717189                                                                           -.169016 + 00                                                                         .472341 - 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01                                                                          .193173 - 01                                                                          2.480  -8.830                     OHARA SF01 -3.533                                                                            1.720041                                                                           -.170120 + 00                                                                         .481532 - 01                                                                          .196576 - 01                                                                          2.450  -8.654                     OHARA SF18 -3.531                                                                            1.721328                                                                           -.170537 + 00                                                                         .482918 - 01                                                                          .197161 - 01                                                                          2.449  -8.650                     SCHOTT                                                                              TISF1                                                                              -3.526                                                                            1.673225                                                                           -.172328 + 00                                                                         .488749 - 01                                                                          -.942108 - 02                                                                         -5.188 18.292                     CORN-F                                                                              717295                                                                             -3.512                                                                            1.717235                                                                           -.168890 + 00                                                                         .480868 - 01                                                                          .128491 - 01                                                                          3.742  -13.144                    SCHOTT                                                                              SF53 -3.416                                                                            1.728118                                                                           -.173616 + 00                                                                         .508275 - 01                                                                          .165347 - 01                                                                          3.074  -10.500                    HOYA  FDS2 -3.403                                                                            1.728117                                                                           -.173734 + 00                                                                         .510503 - 01                                                                          .169420 - 01                                                                          3.013  -10.255                    SCHOTT                                                                              LAF9 -3.379                                                                            1.794839                                                                           -.175409 + 00                                                                         .519128 - 01                                                                          .173355 - 01                                                                          2.995  -10.118                    CORN-F                                                                              728284                                                                             -3.370                                                                            1.728119                                                                           -.175210 + 00                                                                         .519861 - 01                                                                          .122161 - 01                                                                          4.256  -14.343                    HOYA  FD10 -3.347                                                                            1.728067                                                                           -.175779 + 00                                                                         .525203 - 01                                                                          .102814 - 01                                                                          5.108  -17.097                    OHARA SF10 -3.331                                                                            1.728065                                                                           -.174981 + 00                                                                         .525319 - 01                                                                          .137174 - 01                                                                          3.830  -12.756                    CORN-F                                                                              740281                                                                             -3.328                                                                            1.739811                                                                           -.176848 + 00                                                                         .531469 - 01                                                                          .104692 - 01                                                                          5.077  -16.892                    CHANCE                                                                              728284                                                                             -3.321                                                                            1.728063                                                                           -.175405 + 00                                                                         .528124 - 01                                                                          .139044 - 01                                                                          3.798  -12.615                    SCHOTT                                                                              SF10 -3.318                                                                            1.728065                                                                           -.175217 + 00                                                                         .528054 - 01                                                                          .127250 - 01                                                                          4.150  -13.770                    SCHOTT                                                                              SF3  -3.317                                                                            1.739812                                                                           -.176532 + 00                                                                         .532265 - 01                                                                          .199391 -  01                                                                         2.669  -8.854                     HOYA  FD3  -3.309                                                                            1.739810                                                                           -.176226 + 00                                                                         .532600 - 01                                                                          .197209 - 01                                                                          2.701  -8.936                     OHARA SF3  -3.283                                                                            1.739809                                                                           -.175936 + 00                                                                         .535928 - 01                                                                          .207575 - 01                                                                          2.582  -8.476                     SCHOTT                                                                              SF54 -3.279                                                                            1.740610                                                                           -.177244 + 00                                                                         .540529 - 01                                                                          .188731 - 01                                                                          2.864  -9.391                     CHANCE                                                                              748278                                                                             -3.259                                                                            1.748224                                                                           -.178804 + 00                                                                         .548708 - 01                                                                          .210530 - 01                                                                          2.606  -8.493                     OHARA SF04 -3.219                                                                            1.750644                                                                           -.179544 + 00                                                                         .557806 - 01                                                                          .178977 - 01                                                                          3.117  -10.032                    HOYA  FD4  -3.219                                                                            1.755001                                                                           -.180410 + 00                                                                         .560502 - 01                                                                          .157490 - 01                                                                          3.559  -11.455                    CHANCE                                                                              755276                                                                             -3.206                                                                            1.755007                                                                           -.180326 + 00                                                                         .562432 - 01                                                                          .196885 - 01                                                                          2.857  -9.159                     CORN-F                                                                              755276                                                                             -3.197                                                                            1.755006                                                                           -.180036 + 00                                                                         .563099 - 01                                                                          .123656 - 01                                                                          4.554  -14.559                    SCHOTT                                                                              SF63 -3.193                                                                            1.748205                                                                           -.179540 + 00                                                                         .562238 - 01                                                                          .191761 - 01                                                                          2.932  -9.363                     SCHOTT                                                                              SF4  -3.192                                                                            1.755004                                                                           -.180351 +  00                                                                        .564958 - 01                                                                          .205532 - 01                                                                          2.749  -8.775                     OHARA SF13 -3.166                                                                            1.740577                                                                           -.178843 + 00                                                                         .564814 - 01                                                                          .132034 - 01                                                                          4.278  -13.545                    OHARA SF4  -3.154                                                                            1.755002                                                                           -.180714 + 00                                                                         .572879 - 01                                                                          .196616 - 01                                                                          2.914  -9.191                     SCHOTT                                                                              SF61 -3.149                                                                            1.750645                                                                           -.180774 + 00                                                                         .574082 - 01                                                                          .195867 - 01                                                                          2.931  -9.229                     CORN-F                                                                              762269                                                                             -3.141                                                                            1.761598                                                                           -.184460 + 00                                                                         .587192 - 01                                                                          .118663 - 01                                                                          4.948  -15.545                    HOYA  FD13 -3.140                                                                            1.740582                                                                           -.179728 + 00                                                                         .572369 - 01                                                                          .210271 - 01                                                                          2.722  -8.547                     SCHOTT                                                                              SF13 -3.128                                                                            1.740578                                                                           -.179992 + 00                                                                         .575388 - 01                                                                          .889124 - 02                                                                          6.471  -20.244                    SCHOTT                                                                              TIF6 -3.054                                                                            1.616359                                                                           -.160015 + 00                                                                         .523949 - 01                                                                          -.253322 - 01                                                                         -2.068 6.317                      OHARA SFS5 -3.049                                                                            1.761595                                                                           -.183071 + 00                                                                         .600488 - 01                                                                          .170045 - 01                                                                          3.531  -10.766                    HOYA  FDS5 -3.046                                                                            1.761600                                                                           -.184171 + 00                                                                         .604674 - 01                                                                          .131237 - 01                                                                          4.608  -14.033                    SCHOTT                                                                              SF55 -3.040                                                                            1.761596                                                                           -.184320 + 00                                                                         .606329 - 01                                                                          .189400 - 01                                                                          3.201  -9.732                     OHARA SFL14                                                                              -3.030                                                                            1.761615                                                                           -.187228 + 00                                                                         .617896 - 01                                                                          -.149891 - 03                                                                         -412.230                                                                             1249.090                   OHARA SFSL3                                                                              -2.984                                                                            1.784482                                                                           -.188637 + 00                                                                         .632232 - 01                                                                          .241489 - 02                                                                          26.181 -78.114                    CORN-F                                                                              734265                                                                             -2.969                                                                            1.733821                                                                           -.186901 + 00                                                                         .629605 - 01                                                                          -.296774 - 02                                                                         -21.215                                                                              62.978                     CORN-F                                                                              785259                                                                             -2.931                                                                            1.784776                                                                           -.191216 + 00                                                                         .652490 - 01                                                                          .116685 - 01                                                                          5.592  -16.387                    OHARA SF14 -2.917                                                                            1.761613                                                                           -.186558 + 00                                                                         .639638 - 01                                                                          .888133 - 02                                                                          7.202  -21.006                    SCHOTT                                                                              SF14 -2.903                                                                            1.761614                                                                           -.186734 + 00                                                                         .643310 - 01                                                                          .591584 - 02                                                                          10.874 -31.565                    OHARA SFS3 -2.902                                                                            1.784486                                                                           -.188982 + 00                                                                         .651195 - 01                                                                          .178789 - 01                                                                          3.642  -10.570                    SCHOTT                                                                              SF56 -2.895                                                                            1.784485                                                                           -.190232 + 00                                                                         .657154 - 01                                                                          .189545 - 01                                                                          3.467  -10.036                    HOYA  FD14 -2.895                                                                            1.761617                                                                           -.186961 + 00                                                                         .645879 - 01                                                                          .151099 - 01                                                                          4.275  -12.373                    HOYA  FDS3 -2.884                                                                            1.784484                                                                           -.190638 + 00                                                                         .660970 - 01                                                                          .217325 - 01                                                                          3.041  -8.772                     OHARA SFL11                                                                              -2.879                                                                            1.784504                                                                           -.193021 + 00                                                                         .670424 - 01                                                                          .456981 - 03                                                                          146.707                                                                              -422.383                   OHARA SFL6 -2.858                                                                            1.804954                                                                           -.194939 + 00                                                                         .682144 - 01                                                                          .459951 - 02                                                                          14.831 -42.383                    CORN-F                                                                              805255                                                                             -2.841                                                                            1.804867                                                                           -.194059 + 00                                                                         .683118 - 01                                                                          .152341 - 01                                                                          4.484  -12.738                    CHANCE                                                                              805254                                                                             -2.834                                                                            1.804952                                                                           -.195132 + 00                                                                         .688456 - 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01                                                                          .186358 - 01                                                                          3.710  -10.443                    CHANCE                                                                              785258                                                                             -2.810                                                                            1.784501                                                                           -.192403 + 00                                                                         .684671 - 01                                                                          .128753 - 01                                                                          5.318  -14.944                    HOYA  FD6  -2.803                                                                            1.804957                                                                           -.194363 + 00                                                                         .693396 - 01                                                                          .188400 - 01                                                                          3.680  -10.316                    OHARA SF11 -2.796                                                                            1.784502                                                                           -.192467 + 00                                                                         .688348 - 01                                                                          .584361 - 02                                                                          11.780 -32.936                    SCHOTT                                                                              SF11 -2.787                                                                            1.784502                                                                           -.192107 + 00                                                                         .689216 - 01                                                                          .605232 - 02                                                                          11.388 -31.741                    OHARA SF6  -2.779                                                                            1.804950                                                                           -.194546 + 00                                                                         .699951 - 01                                                                          .202121 - 01                                                                          3.463  -9.625                     OHARA LF6  -.559211 + 01                                                                          1.568272                                                                              -.127770 + 00                                                                         .228483 - 01                                                                          -79.861323                        OHARA BAFO3                                                                              -.558236 + 01                                                                          1.686896                                                                              -.124389 + 00                                                                         .222824 - 01                                                                          -79.844049                        CHANCE                                                                              582421                                                                             -.552595 + 01                                                                          1.583140                                                                              -.130084 + 00                                                                         .235406 - 01                                                                          -79.742502                        OHARA LF1  -.545623 + 01                                                                          1.574057                                                                              -.128470 + 00                                                                         .235456 - 01                                                                          -79.614264                        OHARA LF3  -.544102 + 01                                                                          1.583143                                                                              -.129985 + 00                                                                         .238897 - 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01                                                                         -.124346 - 02                                                                         -90.766915                        CHANCE                                                                              524592                                                                             .747616 + 02                                                                           1.524639                                                                              -.933783 - 01                                                                         -.124901 - 02                                                                         -90.766332                        SCHOTT                                                                              LAKN7                                                                              .749037 + 02                                                                           1.652407                                                                              -.944376 - 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02                                                                         -90.717224                        SCHOTT                                                                              SK12 .836981 + 02                                                                           1.583842                                                                              -.929417 - 01                                                                         -.111044 - 02                                                                         -90.684521                        SCHOTT                                                                              BALK1                                                                              .871857 + 02                                                                           1.527057                                                                              -.920710 - 01                                                                         -.105603 - 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01                                                                         -.101877 - 02                                                                         -90.619698                        SCHOTT                                                                              K5   .939519 + 02                                                                           1.523126                                                                              -.929097 - 01                                                                         -.988908 - 03                                                                         -90.609818                        HOYA  ZNC1 .957631 + 02                                                                           1.533814                                                                              -.951004 - 01                                                                         -.993080 - 03                                                                         -90.598286                        OHARA LAK18                                                                              .997389 + 02                                                                           1.730123                                                                              -.101009 + 00                                                                         -.101274 - 02                                                                         -90.574442                        SCHOTT                                                                              BAK2 .103271 + 03                                                                           1.540615                                                                              -.925331 - 01                                                                         -.896020 - 03                                                                         -90.554791                        CORN-F                                                                              604640                                                                             .109857 + 03                                                                           1.604192                                                                              -.863228 - 01                                                                         -.785775 - 03                                                                         -90.521534                        OHARA SK7  .110102 +  03                                                                          1.608032                                                                              -.930232 - 01                                                                         -.844881 - 03                                                                         -90.520373                        SCHOTT                                                                              LAK11                                                                              .110909 + 03                                                                           1.659134                                                                              -.965050 - 01                                                                         -.870126 - 03                                                                         -90.516586                        OHARA LAK02                                                                              .114422 + 03                                                                           1.670846                                                                              -.963556 - 01                                                                         -.842108 - 03                                                                         -90.500729                        OHARA SK3  .116368 + 03                                                                           1.609560                                                                              -.937132 - 01                                                                         -.805320 - 03                                                                         -90.492356                        OHARA PSK02                                                                              .125723 + 03                                                                           1.618706                                                                              -.871342 - 01                                                                         -.693063 - 03                                                                         -90.455719                        HOYA  LAC8 .128627 + 03                                                                           1.713957                                                                              -.102430 + 00                                                                         -.796330 - 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01                                                                         -.447888 - 03                                                                         -90.274261                        CHANCE                                                                              713538                                                                             .211681 + 03                                                                           1.713958                                                                              -.102639 + 00                                                                         -.484875 - 03                                                                         -90.270667                        OHARA LAK9 .222992 + 03                                                                           1.691914                                                                              -.100754 + 00                                                                         -.451830 - 03                                                                         -90.256941                        CORN-F                                                                              623581                                                                             .225515 + 03                                                                           1.623780                                                                              -.951191 - 01                                                                         -.421786 - 03                                                                         -90.254064                        HOYA  ADF1 .228995 + 03                                                                           1.565769                                                                              -.104320 + 00                                                                         -.455555 - 03                                                                         -90.250203                        SCHOTT                                                                              LAK23                                                                              .235819 + 03                                                                           1.669660                                                                              -.962547 - 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03                                                                         -90.222234                        SCHOTT                                                                              LAK8 .280438 + 03                                                                           1.713962                                                                              -.102629 + 00                                                                         -.365959 - 03                                                                         -90.204307                        SCHOTT                                                                              SK4  .306677 + 03                                                                           1.613477                                                                              -.942118 - 01                                                                         -.307202 - 03                                                                         -90.186827                        HOYA  BACD4                                                                              .371294 + 03                                                                           1.613475                                                                              -.942747 - 01                                                                         -.253908 - 03                                                                         -90.154312                        HOYA  BACD3                                                                              .416634 + 03                                                                           1.609561                                                                              -.938263 - 01                                                                         -.225201 - 03                                                                         -90.137521                        CHANCE                                                                              613586                                                                             .428948 + 03                                                                           1.613475                                                                              -.942101 - 01                                                                         -.219630 - 03                                                                         -90.133572                        CHANCE                                                                              658573                                                                             .484001 + 03                                                                           1.659131                                                                              -.964698 - 01                                                                         -.199317 - 03                                                                         -90.118379                        OHARA K3   .569286 + 03                                                                           1.518865                                                                              -.937249 - 01                                                                         -.164636 - 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01                                                                         -.823742 - 05                                                                         -90.004910                        __________________________________________________________________________

                                      TABLE II                                    __________________________________________________________________________    LEAST SQUARES FITTED COEFFICIENTS                                                     ORDER                                                                              BASE WVL              MIN WVL                                                                             MAX WVL                                      3.0  0.5890                0.360 1.014                                CAT   NAME  ETA1/ETA2                                                                             INDEX                                                                              ETA1    ETA2    ETA3    ETA2/ETA3                                                                            ETA1/ETA3             __________________________________________________________________________    CORN-F                                                                              558525                                                                              -1050.530                                                                             1.557434                                                                           -.974752 - 01                                                                         .927867 - 04                                                                          -.390622 - 02                                                                         -.024  24.954                SCHOTT                                                                              LASF11                                                                              -752.430                                                                              1.801525                                                                           -.115759 + 00                                                                         .153847 - 03                                                                          .524469 - 02                                                                          .029   -22.072               OHARA BAF5  -594.102                                                                              1.607198                                                                           -.104136 + 00                                                                         .175283 - 03                                                                          .554173 - 02                                                                          .032   -18.791               SCHOTT                                                                              BAFN11                                                                              -377.046                                                                              1.666620                                                                           -.105738 + 00                                                                         .280437 - 03                                                                          .413480 - 02                                                                          .068   -25.573               HOYA  LAFL4 -351.383                                                                              1.712584                                                                           -.118572 + 00                                                                         .337443 - 03                                                                          .277293 - 02                                                                          .122   -42.760               HOYA  BAF10 -344.527                                                                              1.669924                                                                           -.108569 + 00                                                                         .315124 - 03                                                                          .207762 - 02                                                                          .152   -52.256               OHARA SSK5  -323.740                                                                              1.658342                                                                           -.100614 + 00                                                                         .310787 - 03                                                                          .104009 - 01                                                                          .030   -9.674                SCHOTT                                                                              BAF5  -263.952                                                                              1.607199                                                                           -.103629 + 00                                                                         .392606 - 03                                                                          .601944 - 02                                                                          .065   -17.216               SCHOTT                                                                              LAF23 -245.436                                                                              1.688900                                                                           -.103472 + 00                                                                         .421584 - 03                                                                          .795724 - 02                                                                          .053   -13.003               HOYA  BAF6  -234.257                                                                              1.588914                                                                           -.105634 + 00                                                                         .450934 - 03                                                                          .482195 - 02                                                                          .094   -21.907               SCHOTT                                                                              LAF20 -231.441                                                                              1.682372                                                                           -.106185 + 00                                                                         .458798 - 03                                                                          .612522 - 02                                                                          .075   -17.336               HOYA  NBFD11                                                                              -231.215                                                                              1.785765                                                                           -.116614 + 00                                                                         .504353 - 03                                                                          .718313 - 02                                                                          .070   -16.234               OHARA BAF2  -203.299                                                                              1.569565                                                                           -.103725 + 00                                                                         .510213 - 03                                                                          .582092 - 02                                                                          .088   -17.819               HOYA  BAF5  -189.571                                                                              1.607202                                                                           -.103909 + 00                                                                         .548126 - 03                                                                          .549668 - 02                                                                          .100   -18.904               CORN-F                                                                              548457                                                                              -171.777                                                                              1.548010                                                                           -.112116 + 00                                                                         .652683 - 03                                                                          .238165 - 02                                                                          .274   -47.075               OHARA LAF01 -154.399                                                                              1.699891                                                                           -.106472 + 00                                                                         .689593 - 03                                                                          .486976 - 02                                                                          .142   -21.864               OHARA LAF09 -139.242                                                                              1.696897                                                                           -.105459 + 00                                                                         .757380 -  03                                                                         .640845 - 02                                                                          .118   -16.456               SCHOTT                                                                              LAF10 -118.034                                                                              1.784300                                                                           -.116661 + 00                                                                         .988369 - 03                                                                          .515218 - 02                                                                          .192   -22.643               SCHOTT                                                                              BAF53 -112.012                                                                              1.669925                                                                           -.108658 + 00                                                                         .970051 - 03                                                                          .386597 - 02                                                                          .251   -28.106               CHANCE                                                                              548456                                                                              -107.436                                                                              1.547602                                                                           -.112271 + 00                                                                         .104501 - 02                                                                          .241302 - 02                                                                          .433   -46.527               OHARA BAF6  -100.221                                                                              1.588912                                                                           -.105204 + 00                                                                         .104972 - 02                                                                          .566316 - 02                                                                          .185   -18.577               HOYA  LAFL3 -97.646 1.700195                                                                           -.106981 + 00                                                                         .109560 - 02                                                                          .639820 - 02                                                                          .171   -16.720               HOYA  FEL3  -89.849 1.560038                                                                           -.108708 + 00                                                                         .120990 - 02                                                                          .310242 - 02                                                                          .390   -35.040               HOYA  BAF21 -89.124 1.664119                                                                           -.104787 + 00                                                                         .117575 - 02                                                                          .578126 - 02                                                                          .203   -18.125               OHARA LAF3  -81.326 1.716894                                                                           -.106686 + 00                                                                         .131184 - 02                                                                          .806100 - 02                                                                          .163   -13.235               OHARA BAF1  -80.823 1.556816                                                                           -.105401 + 00                                                                         .130410 - 02                                                                          .509190 - 02                                                                          .256   -20.700               OHARA BAF11 - 77.909                                                                              1.666617                                                                           -.105827 + 00                                                                         .135835 - 02                                                                          .544010 - 02                                                                          .250   -19.453               SCHOTT                                                                              LLF7  -69.953 1.548743                                                                           -.112723 + 00                                                                         .161141 - 02                                                                          .233046 - 02                                                                          .691   -48.369               OHARA LLF3  -69.160 1.560042                                                                           -.108884 + 00                                                                         .157438 - 02                                                                          .372003 - 02                                                                          .423   -29.270               HOYA  BAF9  -68.508 1.643179                                                                           -.106698 + 00                                                                         .155745 - 02                                                                          .564904 - 02                                                                          .276   -18.888               SCHOTT                                                                              KZFSN5                                                                              -65.662 1.653997                                                                           -.129255 + 00                                                                         .196849 - 02                                                                          -.310441 - 02                                                                         -.634  41.636                SCHOTT                                                                              LLF1  -63.655 1.548053                                                                           -.111850 + 00                                                                         .175712 - 02                                                                          .253850 - 02                                                                          .692   -44.061               OHARA LASF01                                                                              -63.641 1.785766                                                                           -.115810 + 00                                                                         .181975 - 02                                                                          .673988 - 02                                                                          .270   -17.183               SCHOTT                                                                              LAFN3 -60.968 1.716888                                                                           -.106539 + 00                                                                         .174745 - 02                                                                          .955062 - 02                                                                          .183   -11.155               CORN-F                                                                              670471                                                                              -58.527 1.669901                                                                           -.108558 + 00                                                                         .185484 - 02                                                                          .814704 - 02                                                                          .228   -13.325               HOYA  FF2   -57.558 1.532476                                                                           -.111413 + 00                                                                         .193568 -  02                                                                         -.109276 - 01                                                                         -.177  10.196                SCHOTT                                                                              BAFN10                                                                              -54.216 1.669927                                                                           -.108512 + 00                                                                         .200146 - 02                                                                          .413130 - 02                                                                          .484   -26.266               OHARA LASF06                                                                              -53.532 1.840854                                                                           -.118293 + 00                                                                         .220977 - 02                                                                          .901010 - 02                                                                          .245   -13.129               OHARA LLF1  -53.174 1.548053                                                                           -.111719 + 00                                                                         .210102 - 02                                                                          .330115 - 02                                                                          .636   -33.843               OHARA LLF7  -52.684 1.548601                                                                           -.112212 + 00                                                                         .212992 - 02                                                                          .302928 - 02                                                                          .703   -37.042               HOYA  FEL1  -51.632 1.548049                                                                           -.111508 + 00                                                                         .215968 - 02                                                                          .364425 - 02                                                                          .593   -30.598               SCHOTT                                                                              TIF2  -51.036 1.532474                                                                           -.111177 + 00                                                                         .217838 - 02                                                                          -.947587 - 02                                                                         -.230  11.733                HOYA  LAF2  -49.984 1.743876                                                                           -.114058 + 00                                                                         .228190 - 02                                                                          .341337 - 02                                                                          .669   -33.415               HOYA  TAFD5 -48.089 1.834857                                                                           -.119028 + 00                                                                         .247515 - 02                                                                          .924471 - 02                                                                          .268   -12.875               SCHOTT                                                                              BAF9  -45.451 1.643179                                                                           -.106557 + 00                                                                         .234445 - 02                                                                          .706099 - 02                                                                          .332   -15.091               OHARA KZFS5 -44.759 1.653994                                                                           -.128928 + 00                                                                         .288054 - 02                                                                          -.161015 - 02                                                                         -1.789 80.072                HOYA  FEL7  -44.140 1.548741                                                                           -.112447 + 00                                                                         .254752 - 02                                                                          .376192 - 02                                                                          .677   -29.891               OHARA BAF9  -40.296 1.643181                                                                           -.106837 + 00                                                                         .265129 - 02                                                                          .686698 - 02                                                                          .386   -15.558               OHARA BAF10 -39.937 1.669925                                                                           -.108092 + 00                                                                         .270654 - 02                                                                          .493908 - 02                                                                          .548   -21.885               OHARA LASF05                                                                              -39.536 1.834664                                                                           -.119657 + 00                                                                         .302656 - 02                                                                          .845778 - 02                                                                          .358   -14.148               SCHOTT                                                                              LLF4  -37.349 1.561290                                                                           -.113002 + 00                                                                         .302554 - 02                                                                          .337006 - 02                                                                          .898   -33.531               HOYA  LAFL1 -36.213 1.685902                                                                           -.103528 + 00                                                                         .285890 - 02                                                                          .119049 - 01                                                                          .240   -8.696                HOYA  NBFD1 -34.398 1.788869                                                                           -.119434 + 00                                                                         .347212 - 02                                                                          .725389 - 02                                                                          .479   -16.465               HOYA  BAF11 -34.301 1.666619                                                                           -.105769 + 00                                                                         .308354 - 02                                                                          .795536 - 02                                                                          .388   -13.295               HOYA  BAF8  -33.837 1.623642                                                                           -.108554 + 00                                                                         .320811 - 02                                                                          .707055 - 02                                                                          .454   -15.353               HOYA  FEL4  -33.462 1.561288                                                                           -.113039 + 00                                                                         .337814 - 02                                                                          .462977 - 02                                                                          .730   -24.416               OHARA LLF8  -32.139 1.532480                                                                           -.111282 + 00                                                                         .346255 - 02                                                                          -.794041 - 02                                                                         -.436  14.015                OHARA LLF4  -30.955 1.561288                                                                           -.113115 + 00                                                                         .365418 - 02                                                                          .425506 - 02                                                                          .859   -26.584               SCHOTT                                                                              BAF8  -29.261 1.623643                                                                           -.108607 + 00                                                                         .371162 - 02                                                                          .742686 - 02                                                                          .500   -14.623               OHARA BAF8  -28.362 1.623643                                                                           -.108774 + 00                                                                         .383526 - 02                                                                          .718593 - 02                                                                          .534   -15.137               SCHOTT                                                                              BAF3  -27.953 1.582578                                                                           -.109864 + 00                                                                         .393029 - 02                                                                          .556866 - 02                                                                          .598   -16.725               CORN-F                                                                              667484                                                                              -26.737 1.666597                                                                           -.105346 + 00                                                                         .394003 - 02                                                                          .112015 - 01                                                                          .352   -9.405                HOYA  BAF3  -24.490 1.582578                                                                           -.109451 + 00                                                                         .446919 - 02                                                                          .820470 - 02                                                                          .545   -13.340               HOYA  BAF7  -24.292 1.607917                                                                           -.110450 + 00                                                                         .454684 - 02                                                                          .674832 - 02                                                                          .674   -16.367               SCHOTT                                                                              BAF52 -23.590 1.608493                                                                           -.109821 + 00                                                                         .465544 - 02                                                                          .636994 - 02                                                                          .731   -17.240               CORN-F                                                                              606439                                                                              -23.512 1.605508                                                                           -.116186 + 00                                                                         .494161 - 02                                                                          .131351 - 02                                                                          3.762  -88.454               HOYA  NBFD12                                                                              -23.111 1.799364                                                                           -.120496 + 00                                                                         .521386 - 02                                                                          .102631 - 01                                                                          .508   -11.741               HOYA  TAFD2 -22.942 1.837551                                                                           -.121732 + 00                                                                         .530614 - 02                                                                          .118741 - 01                                                                          .447   -10.252               OHARA LAF2  -22.723 1.743877                                                                           -.114027 + 00                                                                         .501816 - 02                                                                          .900490 - 02                                                                          .557   -12.663               OHARA BAF3  -22.658 1.582579                                                                           -.110049 + 00                                                                         .485696 - 02                                                                          .780848 - 02                                                                          .622   -14.094               OHARA LASF02                                                                              -22.164 1.799377                                                                           -.120641 + 00                                                                         .544307 - 02                                                                          .803180 - 02                                                                          .678   -15.020               OHARA BAF01 -20.498 1.682384                                                                           -.113645 + 00                                                                         .554431 - 02                                                                          -.319651 - 02                                                                         -1.734 35.553                SCHOTT                                                                              BAF12 -19.769 1.639197                                                                           -.112813 + 00                                                                         .570667 - 02                                                                          .570551 - 02                                                                          1.000  -19.773               SCHOTT                                                                              BASF52                                                                              -19.185 1.701684                                                                           -.124359 + 00                                                                         .648195 - 02                                                                          .301104 - 02                                                                          2.153  -41.301               SCHOTT                                                                              BAF51 -19.184 1.652135                                                                           -.113428 + 00                                                                         .591270 - 02                                                                          .806229 - 02                                                                          .733   -14.069               SCHOTT                                                                              LF8   -19.124 1.564341                                                                           -.116535 + 00                                                                         .609355 - 02                                                                          .539743 - 02                                                                          1.129  -21.591               SCHOTT                                                                              BAF50 -19.106 1.682613                                                                           -.114474 + 00                                                                         .599166 - 02                                                                          .852320 - 02                                                                          .703   -13.431               HOYA  BAF13 -18.841 1.668807                                                                           -.113565 + 00                                                                         .602752 - 02                                                                          .568205 - 02                                                                          1.061  -19.987               HOYA  FL6   -18.794 1.567227                                                                           -.119041 + 00                                                                         .633390 - 02                                                                          .410144 - 02                                                                          1.544  -29.024               HOYA  LAFL5 -18.320 1.719877                                                                           -.115979 + 00                                                                         .633086 - 02                                                                          .685298 - 02                                                                          .924   -16.924               OHARA LF8   -18.227 1.564346                                                                           -.116391 + 00                                                                         .638568 - 02                                                                          .576779 - 02                                                                          1.107  -20.180               SCHOTT                                                                              BAF13 -17.549 1.668811                                                                           -.113273 + 00                                                                         .645484 - 02                                                                          .633754 - 02                                                                          1.019  -17.873               OHARA LAF011                                                                              -17.479 1.719887                                                                           -.110611 + 00                                                                         .632831 - 02                                                                          .120600 - 01                                                                          .525   -9.172                SCHOTT                                                                              LAFN2 -17.470 1.743879                                                                           -.113731 + 00                                                                         .651028 - 02                                                                          .113526 - 01                                                                          .573   -10.018               SCHOTT                                                                              LAF25 -17.388 1.784133                                                                           -.123364 + 00                                                                         .709495 - 02                                                                          .450414 - 02                                                                          1.575  -27.389               HOYA  BAF22 -17.185 1.682384                                                                           -.114314 + 00                                                                         .665199 - 02                                                                          .665086 - 02                                                                          1.000  -17.188               OHARA BAF13 -16.928 1.668813                                                                           -.113221 + 00                                                                         .668836 - 02                                                                          .660533 - 02                                                                          1.013  -17.141               OHARA LASF07                                                                              -16.700 1.862845                                                                           -.122676 + 00                                                                         .734572 - 02                                                                          .118988 - 01                                                                          .617   -10.310               SCHOTT                                                                              LF6   -16.551 1.567223                                                                           -.118934 + 00                                                                         .718604 - 02                                                                          .544410 - 02                                                                          1.320  -21.846               CHANCE                                                                              567428                                                                              -16.328 1.567221                                                                           -.118977 + 00                                                                         .728671 - 02                                                                          .571840 - 02                                                                          1.274  -20.806               OHARA BAF03 -16.241 1.685664                                                                           -.115724 + 00                                                                         .712537 - 02                                                                          .376098 - 02                                                                          1.895  -30.770               HOYA  LAFL6 -16.040 1.719874                                                                           -.120916 + 00                                                                         .753820 - 02                                                                          .637544 - 02                                                                          1.182  -18.966               SCHOTT                                                                              LAFN8 -15.804 1.735068                                                                           -.122469 + 00                                                                         .774926 -  02                                                                         .579672 - 02                                                                          1.337  -21.127               HOYA  BAF20 -15.734 1.685663                                                                           -.115826 + 00                                                                         .736131 - 02                                                                          .575338 - 02                                                                          1.279  -20.132               HOYA  BAF12 -15.690 1.639198                                                                           -.112862 + 00                                                                         .719328 - 02                                                                          .669007 - 02                                                                          1.075  -16.870               HOYA  NBFD2 -15.464 1.797057                                                                           -.123831 + 00                                                                         .800752 - 02                                                                          .109175 - 01                                                                          .733   -11.342               HOYA  FL4   -15.210 1.578346                                                                           -.121940 + 00                                                                         .801715 - 02                                                                          .442920 - 02                                                                          1.810  -27.531               OHARA BAF12 -14.896 1.639196                                                                           -.113404 + 00                                                                         .761280 - 02                                                                          .909745 - 02                                                                          .837   -12.465               CHANCE                                                                              573426                                                                              -14.766 1.572989                                                                           -.119598 + 00                                                                         .809957 - 02                                                                          .675199 - 02                                                                          1.200  -17.713               SCHOTT                                                                              LAF26 -14.692 1.745837                                                                           -.127243 + 00                                                                         .866046 - 02                                                                          .373026 - 02                                                                          2.322  -34.111               SCHOTT                                                                              LASFN3                                                                              -14.611 1.807865                                                                           -.124911 + 00                                                                         .854927 - 02                                                                          .843706 - 02                                                                          1.013  -14.805               OHARA LASF03                                                                              -14.547 1.805953                                                                           -.124719 + 00                                                                         .857340 - 02                                                                          .849751 - 02                                                                          1.009  -14.677               CHANCE                                                                              582421                                                                              - 14.544                                                                              1.582047                                                                           -.121034 + 00                                                                         .832200 - 02                                                                          .618517 - 02                                                                          1.345  -19.568               SCHOTT                                                                              LF1   -14.420 1.572991                                                                           -.119537 + 00                                                                         .828961 - 02                                                                          .659340 - 02                                                                          1.257  -18.130               OHARA LF6   -14.084 1.567226                                                                           -.118862 + 00                                                                         .843972 - 02                                                                          .734061 - 02                                                                          1.150  -16.192               HOYA  BAFD15                                                                              -14.064 1.701858                                                                           -.126284 + 00                                                                         .897935 - 02                                                                          .846666 - 02                                                                          1.061  -14.916               OHARA LF3   -14.044 1.582050                                                                           -.120905 + 00                                                                         .860920 - 02                                                                          .582093 - 02                                                                          1.479  -20.771               OHARA LAF07 -13.840 1.712728                                                                           -.117774 + 00                                                                         .850955 - 02                                                                          .693729 - 02                                                                          1.227  -16.977               SCHOTT                                                                              LF3   -13.696 1.582050                                                                           -.120955 + 00                                                                         .883166 - 02                                                                          .613708 - 02                                                                          1.439  -19.709               HOYA  NBFD13                                                                              -13.631 1.805957                                                                           -.124868 + 00                                                                         .916079 - 02                                                                          .111698 - 01                                                                          .820   -11.179               OHARA LAF02 -13.622 1.719879                                                                           -.116357 + 00                                                                         .854199 - 02                                                                          .881269 - 02                                                                          .969   -13.203               SCHOTT                                                                              LASF31                                                                              -13.091 1.880513                                                                           -.123721 + 00                                                                         .945100 - 02                                                                          .172923 - 01                                                                          .547   -7.155                CHANCE                                                                              606439                                                                              -13.083 1.605513                                                                           -.115836 + 00                                                                         .885368 - 02                                                                          .978139 - 02                                                                          .905   -11.842               SCHOTT                                                                              LF4   -13.066 1.578349                                                                           -.122313 + 00                                                                         .936105 - 02                                                                          .616887 - 02                                                                          1.517  -19.827               SCHOTT                                                                              BAF4  -12.972 1.605519                                                                           -.115760 + 00                                                                         .892355 - 02                                                                          .960315 - 02                                                                          .929   -12.054               OHARA LF1   -12.902 1.572993                                                                           -.119457 + 00                                                                         .925863 - 02                                                                          .779907 - 02                                                                          1.187  -15.317               SCHOTT                                                                              LF7   -12.737 1.574910                                                                           -.122605 + 00                                                                         .962558 - 02                                                                          .651416 - 02                                                                          1.478  -18.821               HOYA  FL3   -12.685 1.582049                                                                           -.121048 + 00                                                                         .954243 - 02                                                                          .652345 - 02                                                                          1.463  -18.556               HOYA  BAF4  -12.677 1.605519                                                                           -.115915 + 00                                                                         .914392 - 02                                                                          .992630 - 02                                                                          .921   -11.678               SCHOTT                                                                              KZFSN7                                                                              -12.518 1.680506                                                                           -.136829 + 00                                                                         .109307 - 01                                                                          -.137634 - 02                                                                         -7.942 99.416                CORN-F                                                                              696365                                                                              -12.487 1.695850                                                                           -.139493 + 00                                                                         .111714 - 01                                                                          .324357 - 02                                                                          3.444  -43.006               SCHOTT                                                                              TIF3  - 12.443                                                                              1.547555                                                                           -.120418 + 00                                                                         .967782 - 02                                                                          -.726416 - 02                                                                         -1.332 16.577                OHARA LASF13                                                                              -12.438 1.804250                                                                           -.128526 + 00                                                                         .103335 - 01                                                                          .786269 - 02                                                                          1.314  -16.346               HOYA  FL1   -12.431 1.572992                                                                           -.119456 + 00                                                                         .960991 - 02                                                                          .818230 - 02                                                                          1.174  -14.599               OHARA FL7   -12.209 1.574905                                                                           -.122511 + 00                                                                         .100343 - 01                                                                          .729838 - 02                                                                          1.375  -16.786               OHARA LASF08                                                                              -12.208 1.882839                                                                           -.124656 + 00                                                                         .102113 - 01                                                                          .165761 - 01                                                                          .616   -7.520                HOYA  FL7   -11.928 1.574909                                                                           -.122437 + 00                                                                         .102643 - 01                                                                          .792764 - 02                                                                          1.295  -15.444               CHANCE                                                                              700412                                                                              -11.906 1.699880                                                                           -.123488 + 00                                                                         .103720 - 01                                                                          .760763 - 02                                                                          1.363  -16.232               OHARA BAF4  -11.837 1.605519                                                                           -.115199 + 00                                                                         .981703 - 02                                                                          .984560 - 02                                                                          .997   -11.802               SCHOTT                                                                              LASF15                                                                              -11.816 1.877832                                                                           -.133184 + 00                                                                         .112711 - 01                                                                          .102777 - 01                                                                          1.097  -12.959               SCHOTT                                                                              BASF57                                                                              -11.777 1.651362                                                                           -.121285 + 00                                                                         .102987 -  01                                                                         .661943 - 02                                                                          1.556  -18.322               HOYA  NBFD3 -11.652 1.804355                                                                           -.128259 + 00                                                                         .110078 - 01                                                                          .119855 - 01                                                                          .918   -10.701               HOYA  BAFD5 -11.534 1.603128                                                                           -.119932 + 00                                                                         .103984 - 01                                                                          .846102 - 02                                                                          1.229  -14.175               OHARA LF4   -11.473 1.578349                                                                           -.122479 + 00                                                                         .106757 - 01                                                                          .766185 - 02                                                                          1.393  -15.986               CHANCE                                                                              581409                                                                              -11.410 1.581340                                                                           -.124300 + 00                                                                         .108938 - 01                                                                          .764572 - 02                                                                          1.425  -16.257               SCHOTT                                                                              BASF5 -11.378 1.603127                                                                           -.119614 + 00                                                                         .105128 - 01                                                                          .927692 - 02                                                                          1.133  -12.894               CORN-F                                                                              581408                                                                              -11.181 1.581292                                                                           -.124582 + 00                                                                         .111423 - 01                                                                          .857334 - 02                                                                          1.300  -14.531               HOYA  FL2   -11.000 1.589106                                                                           -.123975 + 00                                                                         .112707 - 01                                                                          .791513 - 02                                                                          1.424  -15.663               OHARA LF5   -10.756 1.581335                                                                           -.124689 + 00                                                                         .115920 - 01                                                                          .795933 - 02                                                                          1.456  -15.666               OHARA LAF08 -10.587 1.719874                                                                           -.120601 + 00                                                                         .113914 - 01                                                                          .844621 - 02                                                                          1.349  -14.279               OHARA BASF5 - 10.543                                                                              1.603126                                                                           -.119991 + 00                                                                         .113807 - 01                                                                          .105922 - 01                                                                          1.074  -11.328               SCHOTT                                                                              LF5   -10.512 1.581336                                                                           -.124312 + 00                                                                         .118254 - 01                                                                          .854312 - 02                                                                          1.384  -14.551               HOYA  FL5   -10.500 1.581333                                                                           -.124247 + 00                                                                         .118332 - 01                                                                          .915481 - 02                                                                          1.293  -13.572               SCHOTT                                                                              LF2   -10.439 1.589107                                                                           -.124023 + 00                                                                         .118811 - 01                                                                          .819465 - 02                                                                          1.450  -15.135               OHARA LAF8  -10.387 1.735068                                                                           -.123420 + 00                                                                         .118826 - 01                                                                          .865996 - 02                                                                          1.372  -14.252               OHARA LF2   -10.360 1.589102                                                                           -.123610 + 00                                                                         .119316 - 01                                                                          .894944 - 02                                                                          1.333  -13.812               OHARA BASF07                                                                              -10.064 1.701415                                                                           -.123199 + 00                                                                         .122419 - 01                                                                          .876093 - 02                                                                          1.397  -14.062               CORN-F                                                                              702411                                                                              -10.014 1.701680                                                                           -.123468 + 00                                                                         .123294 - 01                                                                          .109797 - 01                                                                          1.123  -11.245               SCHOTT                                                                              BASF6 -9.784  1.667435                                                                           -.120949 + 00                                                                         .123619 - 01                                                                          .912665 - 02                                                                          1.354  -13.252               CORN-F                                                                              865401                                                                              -9.744  1.864842                                                                           -.126497 + 00                                                                         .129823 - 01                                                                          .168720 - 01                                                                          .769   -7.497                HOYA  BAFD6 -9.686  1.667434                                                                           -.121015 + 00                                                                         .124935 - 01                                                                          .954491 - 02                                                                          1.309  -12.679               OHARA BASF7 -9.469  1.701412                                                                           -.122947 + 00                                                                         .129837 - 01                                                                          .578976 - 02                                                                          2.243  -21.235               OHARA BASF6 -9.396  1.667435                                                                           -.120931 + 00                                                                         .128704 - 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01                                                                          .712757 - 02                                                                          2.722  -20.456               CORN-F                                                                              604381                                                                              -7.498  1.603383                                                                           -.132845 + 00                                                                         .177178 - 01                                                                          .109889 - 01                                                                          1.612  -12.089               HOYA  F15   -7.455  1.605534                                                                           -.133509 + 00                                                                         .179078 - 01                                                                          .110112 - 01                                                                          1.626  -12.125               SCHOTT                                                                              F15   -7.303  1.605535                                                                           -.133736 + 00                                                                         .183130 - 01                                                                          .113792 - 01                                                                          1.609  -11.753               SCHOTT                                                                              LAFN7 -7.270  1.749346                                                                           -.144760 + 00                                                                         .199107 - 01                                                                          .758794 - 02                                                                          2.624  -19.078               OHARA F5    -7.206  1.603304                                                                           -.133048 + 00                                                                         .184644 - 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 01                                                                         1.772  -12.674               HOYA  BAFD10                                                                              -7.114  1.650037                                                                           -.128408 + 00                                                                         .180496 - 01                                                                          .129905 - 01                                                                          1.389  -9.885                OHARA BASF12                                                                              -7.018  1.669856                                                                           -.128397 + 00                                                                         .182949 - 01                                                                          .101700 - 01                                                                          1.799  -12.625               SCHOTT                                                                              BASF1 -7.013  1.625944                                                                           -.129741 + 00                                                                         .185012 - 01                                                                          .124187 - 01                                                                          1.490  -10.447               HOYA  NBFD7 -6.998  1.782841                                                                           -.140110 + 00                                                                         .200213 - 01                                                                          .234138 - 02                                                                          8.551  -59.841               SCHOTT                                                                              BASF13                                                                              -6.990  1.697479                                                                           -.130948 + 00                                                                         .187347 - 01                                                                          .716289 - 02                                                                          2.616  -18.281               OHARA BASF1 -6.879  1.625941                                                                           -.129235 + 00                                                                         .187870 - 01                                                                          .135298 - 01                                                                          1.389  -9.552                OHARA BASF08                                                                              -6.850  1.723282                                                                           -.132830 + 00                                                                         .193904 - 01                                                                          .135401 - 01                                                                          1.432  -9.810                SCHOTT                                                                              FN11  -6.827  1.620837                                                                           -.139550 + 00                                                                         .204406 - 01                                                                          -.595771 - 02                                                                         -3.431 23.423                SCHOTT                                                                              F3    -6.777  1.612811                                                                           -.136443 + 00                                                                         .201321 - 01                                                                          .118564 - 01                                                                          1.698  -11.508               HOYA  BAFD8 -6.767  1.723278                                                                           - .132746 + 00                                                                        .196157 - 01                                                                          .593488 - 02                                                                          3.305  -22.367               HOYA  F11   -6.766  1.620837                                                                           -.140410 + 00                                                                         .207514 - 01                                                                          -.576843 - 02                                                                         -3.597 24.341                OHARA F11   -6.751  1.620830                                                                           -.140703 + 00                                                                         .208427 - 01                                                                          -.546265 - 02                                                                         -3.815 25.757                HOYA  F3    -6.670  1.612808                                                                           -.136691 + 00                                                                         .204946 - 01                                                                          .116985 - 01                                                                          1.752  -11.684               SCHOTT                                                                              BASF10                                                                              -6.667  1.650040                                                                           -.128991 + 00                                                                         .193478 - 01                                                                          .144917 - 01                                                                          1.335  -8.901                SCHOTT                                                                              LAF22 -6.643  1.781638                                                                           -.136077 + 00                                                                         .204842 - 01                                                                          .762574 - 02                                                                          2.686  -17.845               SCHOTT                                                                              LASF13                                                                              -6.603  1.855273                                                                           -.137989 + 00                                                                         .208967 - 01                                                                          .169617 - 01                                                                          1.232  -8.135                OHARA F3    -6.561  1.612807                                                                           -.136506 + 00                                                                         .208068 - 01                                                                          .127433 - 01                                                                          1.633  -10.712               OHARA BASF10                                                                              -6.551  1.650038                                                                           -.128157 + 00                                                                         .195643 - 01                                                                          .149071 - 01                                                                          1.312  -8.597                HOYA  BAFD4 -6.550  1.651154                                                                           -.131869 + 00                                                                         .201337 - 01                                                                          .143726 - 01                                                                          1.401  -9.175                OHARA BASF8 -6.525  1.723282                                                                           -.133052 + 00                                                                         .203923 - 01                                                                          .807924 - 02                                                                          2.524  -16.468               HOYA  BAFD1 -6.508  1.625941                                                                           -.128946 + 00                                                                         .198145 - 01                                                                          .143214 - 01                                                                          1.384  -9.004                SCHOTT                                                                              F4    -6.456  1.616469                                                                           -.137862 + 00                                                                         .213540 - 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01                                                                          .131183 - 01                                                                          1.700  -10.580               SCHOTT                                                                              TIF4  -6.214  1.583949                                                                           -.136114 + 00                                                                         .219042 - 01                                                                          -.423007 - 02                                                                         -5.178 32.178                CORN-F                                                                              620363                                                                              -6.144  1.619886                                                                           -.139062 + 00                                                                         .226337 - 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01                                                                          .108494 - 01                                                                          2.277  -13.059               OHARA LAFO12                                                                              -5.701  1.782843                                                                           -.139314 + 00                                                                         .244377 - 01                                                                          .703459 - 02                                                                          3.474  -19.804               CORN-F                                                                              626356                                                                              -5.697  1.625775                                                                           -.141656 + 00                                                                         .248670 - 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01                                                                          -.388728 - 02                                                                         -6.477 36.468                HOYA  FF5   -5.598  1.592582                                                                           -.142075 + 00                                                                         .253793 - 01                                                                          -.249450 - 02                                                                         -10.174                                                                              56.955                SCHOTT                                                                              BASF55                                                                              -5.591  1.699664                                                                           -.145113 + 00                                                                         .259548 - 01                                                                          .348875 - 02                                                                          7.440  -41.595               SCHOTT                                                                              LASF18                                                                              -5.578  1.913275                                                                           -.155052 + 00                                                                         .277982 - 01                                                                          .999495 - 02                                                                          2.781  -15.513               OHARA F6    - 5.565 1.636227                                                                           -.142394 + 00                                                                         .255880 - 01                                                                          .140436 - 01                                                                          1.822  -10.140               OHARA F16   -5.563  1.592579                                                                           -.141625 + 00                                                                         .254589 - 01                                                                          -.133843 - 02                                                                         -19.021                                                                              105.814               HOYA  F6    -5.509  1.636225                                                                           -.142381 + 00                                                                         .258436 - 01                                                                          .145004 - 01                                                                          1.782  -9.819                SCHOTT                                                                              F6    -5.486  1.636228                                                                           -.142495 + 00                                                                         .259741 - 01                                                                          .143701 - 01                                                                          1.808  -9.916                SCHOTT                                                                              F7    -5.457  1.625233                                                                           -.141604 + 00                                                                         .259511 - 01                                                                          .143641 - 01                                                                          1.807  -9.858                HOYA  F7    -5.419  1.625235                                                                           -.141451 + 00                                                                         .261046 - 01                                                                          .146792 - 01                                                                          1.778  -9.636                OHARA LAF7  -5.279  1.749342                                                                           -.142977 + 00                                                                         .270864 - 01                                                                          .944421 - 02                                                                          2.868  -15.139               SCHOTT                                                                              SF7   -5.274  1.639667                                                                           -.145428 + 00                                                                         .275734 - 01                                                                          .145731 - 01                                                                          1.892  -9.979                SCHOTT                                                                              BASF2 -5.257  1.664325                                                                           -.140417 + 00                                                                         .267122 - 01                                                                          .144307 - 01                                                                          1.851  -9.730                HOYA  FD7   -5.233  1.639669                                                                           -.145636 + 00                                                                         .278294 -  01                                                                         .147764 - 01                                                                          1.883  -9.856                OHARA SF7   -5.213  1.639664                                                                           -.145486 + 00                                                                         .279095 - 01                                                                          .151326 - 01                                                                          1.844  -9.614                HOYA  NBFD6 -5.193  1.756551                                                                           -.138386 + 00                                                                         .266493 - 01                                                                          .177375 - 01                                                                          1.502  -7.802                SCHOTT                                                                              BASF14                                                                              -5.120  1.699536                                                                           -.143764 + 00                                                                         .280794 - 01                                                                          .905315 - 02                                                                          3.102  -15.880               HOYA  BAFD14                                                                              -5.098  1.699533                                                                           -.143909 + 00                                                                         .282298 - 01                                                                          .956098 - 02                                                                          2.953  -15.052               HOYA  BAFD2 -5.084  1.664323                                                                           -.140014 + 00                                                                         .275421 - 01                                                                          .162454 - 01                                                                          1.695  -8.619                SCHOTT                                                                              SF12  -5.019  1.648171                                                                           -.148531 + 00                                                                         .295908 - 01                                                                          .122520 - 01                                                                          2.415  -12.123               CHANCE                                                                              648338                                                                              -5.010  1.648173                                                                           -.148536 + 00                                                                         .296460 - 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01                                                                          .201143 - 01                                                                          1.492  -7.383                SCHOTT                                                                              SF9   - 4.921 1.654320                                                                           -.149368 + 00                                                                         .303534 - 01                                                                          .154365 - 01                                                                          1.966  -9.676                HOYA  FD2   -4.904  1.647554                                                                           -.148559 + 00                                                                         .302920 - 01                                                                          .146703 - 01                                                                          2.065  -10.127               SCHOTT                                                                              LASF33                                                                              -4.895  1.805788                                                                           -.146754 + 00                                                                         .299778 - 01                                                                          .149144 - 01                                                                          2.010  -9.840                HOYA  FD9   -4.885  1.654321                                                                           -.149021 + 00                                                                         .305049 - 01                                                                          .155978 - 01                                                                          1.956  -9.554                SCHOTT                                                                              SF2   -4.863  1.647550                                                                           -.148460 + 00                                                                         .305265 - 01                                                                          .154254 - 01                                                                          1.979  -9.624                HOYA  FD12  -4.850  1.648169                                                                           -.148675 + 00                                                                         .306537 - 01                                                                          .141316 - 01                                                                          2.169  -10.521               OHARA SF9   -4.845  1.654317                                                                           -.149580 + 00                                                                         .308741 - 01                                                                          .158054 - 01                                                                          1.953  -9.464                OHARA SF2   -4.834  1.647550                                                                           -.148652 + 00                                                                         .307503 - 01                                                                          .163419 - 01                                                                          1.882  -9.096                OHARA BASF2 -4.825  1.664327                                                                           -.140209 + 00                                                                         .290599 - 01                                                                          .172396 - 01                                                                          1.686  -8.133                SCHOTT                                                                              SF17  -4.817  1.650030                                                                           -.149220 + 00                                                                         .309753 - 01                                                                          .155351 - 01                                                                          1.994  -9.605                CHANCE                                                                              653335                                                                              -4.805  1.653335                                                                           -.150134 + 00                                                                         .312471 - 01                                                                          .161343 - 01                                                                          1.937  -9.305                OHARA SF19  -4.736  1.666653                                                                           -.152025 + 00                                                                         .320978 - 01                                                                          .164376 - 01                                                                          1.953  -9.249                OHARA LAFO15                                                                              -4.713  1.749342                                                                           -.143590 + 00                                                                         .304670 - 01                                                                          .171232 - 01                                                                          1.779  -8.386                SCHOTT                                                                              SF19  -4.680  1.666653                                                                           -.152114 + 00                                                                         .325027 - 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01                                                                          .215868 - 01                                                                          3.781  -9.575                SCHOTT                                                                              SF58  -2.235  1.917309                                                                           -.228319 + 00                                                                         .102142 + 00                                                                          .149349 - 01                                                                          6.839  -15.288               OHARA SFS01 -2.208  1.922555                                                                           -.230396 + 00                                                                         .104355 + 00                                                                          .113419 - 01                                                                          9.201  -20.314               OHARA SFS1  -2.128  1.922547                                                                           -.234526 + 00                                                                         .110212 + 00                                                                          .744372 - 02                                                                          14.806 -31.507               SCHOTT                                                                              SF59  -2.109  1.952164                                                                           -.240117 + 00                                                                         .113849 + 00                                                                          .162320 - 02                                                                          70.139 -147.928              OHARA BK9   2.861   1.493821                                                                           -.803643 - 01                                                                         -.280896 - 01                                                                         -.204052 - 01                                                                         1.377  3.938                 SCHOTT                                                                              FK5   2.865   1.487438                                                                           -.753913 - 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01                                                                         -.226923 - 01                                                                         -.143658 - 01                                                                         1.580  6.091                 HOYA  ZNC7  3.900   1.508409                                                                           -.858151 - 01                                                                         -.220015 - 01                                                                         -.132040 - 01                                                                         1.666  6.499                 SCHOTT                                                                              ZKN7  3.914   1.508407                                                                           -.859278 - 01                                                                         -.219515 - 01                                                                         -.135941 - 01                                                                         1.615  6.321                 CORN-F                                                                              510635                                                                              3.945   1.509933                                                                           -.827472 - 01                                                                         -.209770 - 01                                                                         -.123052 - 01                                                                         1.705  6.725                 CHANCE                                                                              508612                                                                              3.952   1.507526                                                                           -.859436 - 01                                                                         -.217458 - 01                                                                         -.137590 - 01                                                                         1.580  6.246                 HOYA  PCD2  4.003   1.568663                                                                           -.831973 - 01                                                                         -.207821 - 01                                                                         -.122812 - 01                                                                         1.692  6.774                 OHARA ZK7   4.006   1.508408                                                                           -.863600 - 01                                                                         -.215569 - 01                                                                         -.125222 - 01                                                                         1.721  6.897                 SCHOTT                                                                              PSK3  4.017   1.552257                                                                           -.827823 - 01                                                                         -.206068 - 01                                                                         -.119426 - 01                                                                         1.725  6.932                 HOYA  BSC1  4.059   1.510031                                                                           -.827665 - 01                                                                         -.203921 - 01                                                                         -.120206 - 01                                                                         1.696  6.885                 SCHOTT                                                                              PSK50 4.060   1.557468                                                                           -.780233 - 01                                                                         -.192169 - 01                                                                         -.105060 - 01                                                                         1.829  7.427                 SCHOTT                                                                              K51   4.098   1.505117                                                                           -.881892 - 01                                                                         -.215198 - 01                                                                         -.151929 - 01                                                                         1.416  5.805                 SCHOTT                                                                              PSK2  4.101   1.568663                                                                           -.832147 - 01                                                                         -.202894 - 01                                                                         -.115953 - 01                                                                         1.750  7.177                 SCHOTT                                                                              BK1   4.108   1.510030                                                                           -.827070 - 01                                                                         -.201308 - 01                                                                         -.119323 - 01                                                                         1.687  6.931                 OHARA PSK2  4.128   1.568662                                                                           -.830341 - 01                                                                         -.201158 - 01                                                                         -.114936 - 01                                                                         1.750  7.224                 SCHOTT                                                                              KZFSN2                                                                              4.129   1.558285                                                                           -.970219 - 01                                                                         -.234949 - 01                                                                         -.187953 - 01                                                                         1.250  5.162                 SCHOTT                                                                              FK54  4.131   1.436964                                                                           -.577929 - 01                                                                         -.139897 - 01                                                                         -.573847 - 02                                                                         2.438  10.071                OHARA BK1   4.179   1.510032                                                                           -.825826 - 01                                                                         -.197592 - 01                                                                         -.103512 - 01                                                                         1.909  7.978                 OHARA PSK1  4.183   1.547646                                                                           -.833594 - 01                                                                         -.199288 - 01                                                                         -.108206 - 01                                                                         1.842  7.704                 HOYA  PCD3  4.295   1.552255                                                                           -.827597 - 01                                                                         -.192709 - 01                                                                         -.992002 - 02                                                                         1.943  8.343                 SCHOTT                                                                              BK6   4.356   1.531066                                                                           -.843429 - 01                                                                         -.193615 - 01                                                                         -.107182 - 01                                                                         1.806  7.869                 CHANCE                                                                              569631                                                                              4.362   1.568661                                                                           -.831007 - 01                                                                         -.190507 - 01                                                                         -.101752 - 01                                                                         1.872  8.167                 OHARA BK6   4.459   1.531062                                                                           -.839051 - 01                                                                         -.188158 - 01                                                                         -.907099 - 02                                                                         2.074  9.250                 SCHOTT                                                                              K11   4.520   1.500069                                                                           -.852812 - 01                                                                         -.188674 - 01                                                                         -.121052 - 01                                                                         1.559  7.045                 CORN-F                                                                              518603                                                                              4.533   1.518137                                                                           -.869053 - 01                                                                         -.191735 - 01                                                                         -.111896 - 01                                                                         1.714  7.767                 OHARA BK2   4.606   1.506507                                                                           -.844496 - 01                                                                         -.183355 - 01                                                                         -.948703 - 02                                                                         1.933  8.902                 CORN-F                                                                              564609                                                                              4.654   1.563936                                                                           -.859088 - 01                                                                         -.184594 - 01                                                                         -.927684 - 02                                                                         1.990  9.261                 HOYA  LACL6 4.704   1.639924                                                                           -.868927 - 01                                                                         -.184710 - 01                                                                         -.862896 - 02                                                                         2.141  10.070                HOYA  BSC6  4.709   1.531064                                                                           -.842204 - 01                                                                         -.178850 - 01                                                                         -.870653 - 02                                                                         2.054  9.673                 CHANCE                                                                              589613                                                                              4.770   1.589057                                                                           -.854452 - 01                                                                         -.179132 - 01                                                                         -.941120 - 02                                                                         1.903  9.079                 CORN-F                                                                              540597                                                                              4.851   1.539438                                                                           -.876151 - 01                                                                         -.180594 - 01                                                                         -.102980 - 01                                                                         1.754  8.508                 SCHOTT                                                                              SK5   4.875   1.589059                                                                           -.853376 - 01                                                                         -.175035 -  01                                                                        -.825204 - 02                                                                         2.121  10.341                OHARA SK20  4.884   1.559557                                                                           -.854291 - 01                                                                         -.174927 - 01                                                                         -.809035 - 02                                                                         2.162  10.559                HOYA  FCD10 4.904   1.456463                                                                           -.575592 - 01                                                                         -.117369 - 01                                                                         -.235336 - 02                                                                         4.987  24.458                SCHOTT                                                                              LAK31 4.909   1.696641                                                                           -.926829 - 01                                                                         -.188813 - 01                                                                         -.103613 - 01                                                                         1.822  8.945                 SCHOTT                                                                              PSK52 4.969   1.603031                                                                           -.798370 - 01                                                                         -.160673 - 01                                                                         -.689272 - 02                                                                         2.331  11.583                OHARA K1    4.980   1.509706                                                                           -.841213 - 01                                                                         -.168921 - 01                                                                         -.744747 - 02                                                                         2.268  11.295                OHARA LAK014                                                                              5.015   1.696705                                                                           -.925209 - 01                                                                         -.184487 - 01                                                                         -.951720 - 02                                                                         1.938  9.721                 HOYA  BACD5 5.026   1.589057                                                                           -.853142 - 01                                                                         -.169741 - 01                                                                         -.765990 - 02                                                                         2.216  11.138                CHANCE                                                                              697562                                                                              5.048   1.697243                                                                           -.929825 - 01                                                                         -.184184 - 01                                                                         -.940164 - 02                                                                         1.959  9.890                 SCHOTT                                                                              UK50  5.057   1.522507                                                                           -.865220 - 01                                                                         -.171107 - 01                                                                         -.854471 - 02                                                                         2.002  10.126                CORN-F                                                                              486817                                                                              5.064   1.486000                                                                           -.638663 - 01                                                                         -.126120 - 01                                                                         -.337675 - 02                                                                         3.735  18.914                CORN-F                                                                              511604                                                                              5.127   1.510937                                                                           -.865685 - 01                                                                         -.168846 - 01                                                                         -.727218 - 02                                                                         2.322  11.904                OHARA SK11  5.140   1.563770                                                                           -.860269 - 01                                                                         -.167378 - 01                                                                         -.734932 - 02                                                                         2.277  11.705                SCHOTT                                                                              K50   5.158   1.522507                                                                           -.867957 - 01                                                                         -.168285 - 01                                                                         -.805276 - 02                                                                         2.090  10.778                SCHOTT                                                                              LAK21 5.183   1.640417                                                                           -.868864 - 01                                                                         -.167639 - 01                                                                         -.758897 - 02                                                                         2.209  11.449                SCHOTT                                                                              FK51  5.195   1.486518                                                                           -.617162 - 01                                                                         -.118794 - 01                                                                         -.295916 - 02                                                                         4.014  20.856                CORN-F                                                                              697562                                                                              5.201   1.696908                                                                           -.929539 - 01                                                                         -.178730 - 01                                                                         -.870714 - 02                                                                         2.053  10.676                CORN-F                                                                              589612                                                                              5.201   1.588927                                                                           -.853444 - 01                                                                         -.164087 - 01                                                                         -.673590 - 02                                                                         2.436  12.670                SCHOTT                                                                              SK20  5.221   1.559563                                                                           -.852772 - 01                                                                         -.163329 - 01                                                                         -.702539 - 02                                                                         2.325  12.138                HOYA  BACD16                                                                              5.244   1.620333                                                                           -.865455 - 01                                                                         -.165044 - 01                                                                         -.727648 - 02                                                                         2.268  11.894                OHARA LAK6  5.246   1.642414                                                                           -.894287 - 01                                                                         -.170461 - 01                                                                         -.800976 - 02                                                                         2.128  11.165                OHARA FK01  5.273   1.496953                                                                           -.639181 - 01                                                                         -.121214 - 01                                                                         -.282021 - 02                                                                         4.298  22.664                OHARA LAK14 5.280   1.696705                                                                           -.939134 - 01                                                                         -.177852 - 01                                                                         -.877056 - 02                                                                         2.028  10.708                HOYA  LACL7 5.283   1.669910                                                                           -.910488 - 01                                                                         -.172354 - 01                                                                         -.690969 - 02                                                                         2.494  13.177                HOYA  ADF1  5.284   1.564918                                                                           -.985658 - 01                                                                         -.186537 - 01                                                                         -.125719 - 01                                                                         1.484  7.840                 CORN-F                                                                              523594                                                                              5.284   1.522931                                                                           -.878077 - 01                                                                         -.166168 - 01                                                                         -.907534 - 02                                                                         1.831  9.675                 CHANCE                                                                              697554                                                                              5.305   1.696705                                                                           -.942321 - 01                                                                         -.177644 - 01                                                                         -.873633 - 02                                                                         2.033  10.786                SCHOTT                                                                              SK16  5.308   1.620335                                                                           -.865128 - 01                                                                         -.162986 - 01                                                                         -.709247 - 02                                                                         2.298  12.198                OHARA LAK7  5.324   1.651515                                                                           -.889115 - 01                                                                         -.166990 - 01                                                                         -.712987 - 02                                                                         2.342  12.470                SCHOTT                                                                              BALKN3                                                                              5.354   1.518425                                                                           -.866198 - 01                                                                         -.161787 - 01                                                                         -.781501 - 02                                                                         2.070  11.084                OHARA K50   5.382   1.522501                                                                           -.869886 - 01                                                                         -.161615 - 01                                                                         -.901797 - 02                                                                         1.792  9.646                 HOYA  LAC14 5.399   1.696709                                                                           -.941316 - 01                                                                         -.174345 - 01                                                                         -.799765 - 02                                                                         2.180  11.770                HOYA  C7    5.425   1.511059                                                                           -.861484 - 01                                                                         -.158813 - 01                                                                         -.577050 - 02                                                                         2.752  14.929                SCHOTT                                                                              TIK1  5.446   1.478630                                                                           -.889197 - 01                                                                         -.163273 - 01                                                                         -.906127 - 02                                                                         1.802  9.813                 SCHOTT                                                                              LAKN14                                                                              5.459   1.696707                                                                           -.941778 - 01                                                                         -.172531 - 01                                                                         -.835559 - 02                                                                         2.065  11.271                CHANCE                                                                              603606                                                                              5.467   1.603035                                                                           -.860735 - 01                                                                         -.157455 - 01                                                                         -.621115 - 02                                                                         2.535  13.858                OHARA SK5   5.472   1.589055                                                                           -.855443 - 01                                                                         -.156340 - 01                                                                         -.591099 - 02                                                                         2.645  14.472                SCHOTT                                                                              K7    5.485   1.511058                                                                           -.863398 - 01                                                                         -.157415 - 01                                                                         -.709100 - 02                                                                         2.220  12.176                HOYA  BACD14                                                                              5.497   1.603037                                                                           -.859069 - 01                                                                         -.156274 - 01                                                                         -.598718 - 02                                                                         2.610  14.349                HOYA  LAC6  5.505   1.642416                                                                           -.899007 - 01                                                                         -.163322 - 01                                                                         -.705862 - 02                                                                         2.314  12.736                SCHOTT                                                                              SK11  5.515   1.563771                                                                           -.857460 - 01                                                                         -.155488 - 01                                                                         -.607616 - 02                                                                         2.559  14.112                CORN-F                                                                              697554                                                                              5.518   1.696912                                                                           -.942449 - 01                                                                         -.170810 - 01                                                                         -.808880 - 02                                                                         2.112  11.651                HOYA  LAC15 5.536   1.696703                                                                           -.930117 - 01                                                                         -.168024 - 01                                                                         -.636116 - 02                                                                         2.641  14.622                SCHOTT                                                                              SK14  5.540   1.603037                                                                           -.860309 - 01                                                                         -.155286 - 01                                                                         -.604468 - 02                                                                         2.569  14.232                OHARA SK14  5.546   1.603039                                                                           -.858583 - 01                                                                         -.154822 - 01                                                                         -.594749 - 02                                                                         2.603  14.436                CORN-F                                                                              641601                                                                              5.546   1.640416                                                                           -.867823 - 01                                                                         -.156484 - 01                                                                         -.606702 - 02                                                                         2.579  14.304                CHANCE                                                                              519604                                                                              5.551   1.518924                                                                           -.862977 - 01                                                                         -.155461 - 01                                                                         -.661172 - 02                                                                         2.351  13.052                CHANCE                                                                              564608                                                                              5.563   1.563770                                                                           -.847603 - 01                                                                         -.154164 - 01                                                                         -.594402 - 02                                                                         2.594  14.428                OHARA BALK3 5.569   1.518287                                                                           -.865144 - 01                                                                         -.155354 - 01                                                                         -.650411 - 02                                                                         2.389  13.301                HOYA  BACD11                                                                              5.580   1.563771                                                                           -.856229 - 01                                                                         -.153440 - 01                                                                         -.627898 - 02                                                                         2.444  13.636                HOYA  BACD165                                                                             5.506   1.620335                                                                           -.870460 - 01                                                                         -.155275 - 01                                                                         -.543135 - 02                                                                         2.859  16.027                CORN-F                                                                              620603                                                                              5.621   1.620330                                                                           -.864614 - 01                                                                         -.153816 - 01                                                                         -.551740 - 02                                                                         2.788  15.671                CHANCE                                                                              620603                                                                              5.698   1.620332                                                                           -.863890 - 01                                                                         -.151603 - 01                                                                         -.556257 - 02                                                                         2.725  15.530                OHARA K7    5.723   1.511058                                                                           -.862441 - 01                                                                         -.150695 - 01                                                                         -.606722 - 02                                                                         2.484  14.215                CORN-F                                                                              652585                                                                              5.741   1.651517                                                                           -.890910 - 01                                                                         -.155172 - 01                                                                         -.540970 - 02                                                                         2.868  16.469                SCHOTT                                                                              FK52  5.757   1.486008                                                                           -.636069 - 01                                                                         -.110490 - 01                                                                         -.176256 - 02                                                                         6.269  36.088                HOYA  LAC7  5.793   1.651516                                                                           -.890710 - 01                                                                         -.153745 - 01                                                                         -.463508 - 02                                                                         3.317  19.217                OHARA PSK01 5.814   1.616928                                                                           -.830255 - 01                                                                         -.142800 - 01                                                                         -.507654 -  02                                                                        2.813  16.355                HOYA  LAC11 5.858   1.658211                                                                           -.909119 - 01                                                                         -.155199 - 01                                                                         -.562700 - 02                                                                         2.758  16.156                OHARA SK16  5.878   1.620334                                                                           -.864307 - 01                                                                         -.147049 - 01                                                                         -.533035 - 02                                                                         2.759  16.215                OHARA BALK1 5.891   1.526355                                                                           -.868165 - 01                                                                         -.147361 - 01                                                                         -.474932 - 02                                                                         3.103  18.280                HOYA  LACL1 5.938   1.640768                                                                           -.915550 - 01                                                                         -.154181 - 01                                                                         -.735562 - 02                                                                         2.096  12.447                HOYA  LAC9  5.942   1.690902                                                                           -.950630 - 01                                                                         -.159979 - 01                                                                         -.704038 - 02                                                                         2.272  13.503                OHARA LAK18 5.942   1.729059                                                                           -.952316 - 01                                                                         -.160259 - 01                                                                         -.688214 - 02                                                                         2.329  13.837                CORN-F                                                                              542734                                                                              5.962   1.542146                                                                           -.708682 - 01                                                                         -.118868 - 01                                                                         -.182265 - 02                                                                         6.522  38.882                HOYA  BACL3 5.999   1.518287                                                                           -.862077 - 01                                                                         -.143706 - 01                                                                         -.403006 - 02                                                                         3.566  21.391                SCHOTT                                                                              K5    6.089   1.522424                                                                           -.875396 - 01                                                                         -.143774 - 01                                                                         -.559524 - 02                                                                         2.570  15.645                SCHOTT                                                                              K4    6.146   1.518884                                                                           -.907123 -  01                                                                        -.147606 - 01                                                                         -.709774 - 02                                                                         2.080  12.781                CHANCE                                                                              713538                                                                              6.147   1.712901                                                                           -.967482 - 01                                                                         -.157384 - 01                                                                         -.675252 - 02                                                                         2.331  14.328                OHARA K5    6.150   1.522429                                                                           -.871256 - 01                                                                         -.141669 - 01                                                                         -.590165 - 02                                                                         2.400  14.763                SCHOTT                                                                              LAK8  6.151   1.712905                                                                           -.967281 - 01                                                                         -.157262 - 01                                                                         -.711774 - 02                                                                         2.209  13.590                HOYA  BACD12                                                                              6.203   1.583057                                                                           -.873666 - 01                                                                         -.140851 - 01                                                                         -.440023 - 02                                                                         3.201  19.855                SCHOTT                                                                              LAKN7 6.213   1.651518                                                                           -.889269 - 01                                                                         -.143121 - 01                                                                         -.481842 - 02                                                                         2.970  18.456                CHANCE                                                                              607595                                                                              6.216   1.607215                                                                           -.875668 - 01                                                                         -.140868 - 01                                                                         -.458911 - 02                                                                         3.070  19.081                OHARA LAK9  6.218   1.690909                                                                           -.949531 - 01                                                                         -.152697 - 01                                                                         -.629513 - 02                                                                         2.426  15.084                CORN-F                                                                              613585                                                                              6.232   1.612616                                                                           -.889349 - 01                                                                         -.142702 - 01                                                                         -.464760 - 02                                                                         3.070  19.136                SCHOTT                                                                              LAK11 6.233   1.658216                                                                           -.909186 - 01                                                                         -.145874 - 01                                                                         -.523021 - 02                                                                         2.789  17.383                SCHOTT                                                                              BAK50 6.237   1.567664                                                                           -.897380 - 01                                                                         -.143880 - 01                                                                         -.501834 - 02                                                                         2.867  17.882                CHANCE                                                                              652585                                                                              6.241   1.651515                                                                           -.889501 - 01                                                                         -.142526 - 01                                                                         -.459356 - 02                                                                         3.103  19.364                SCHOTT                                                                              LAK9  6.277   1.690909                                                                           -.951366 - 01                                                                         -.151571 - 01                                                                         -.622629 - 02                                                                         2.434  15.280                OHARA SK12  6.313   1.583056                                                                           -.875592 - 01                                                                         -.138698 - 01                                                                         -.431570 - 02                                                                         3.214  20.289                CHANCE                                                                              524592                                                                              6.327   1.523932                                                                           -.878863 - 01                                                                         -.138896 - 01                                                                         -.445128 - 02                                                                         3.120  19.744                SCHOTT                                                                              K3    6.336   1.518164                                                                           -.882051 - 01                                                                         -.139217 - 01                                                                         -.551445 - 02                                                                         2.525  15.995                CORN-F                                                                              623581                                                                              6.368   1.622924                                                                           -.896154 - 01                                                                         -.140731 - 01                                                                         -.509693 - 02                                                                         2.761  17.582                HOYA  BACL1 6.376   1.526353                                                                           -.866298 - 01                                                                         -.135861 - 01                                                                         -.445183 - 02                                                                         3.052  19.459                HOYA  SBF5  6.382   1.521226                                                                           -.986168 - 01                                                                         -.154515 - 01                                                                         -.988293 - 02                                                                         1.563  9.978                 SCHOTT                                                                              BALK1 6.383   1.526356                                                                           -.866673 - 01                                                                         -.135777 - 01                                                                         -.425036 - 02                                                                         3.194  20.391                HOYA  LAC8  6.449   1.712902                                                                           -.964279 - 01                                                                         -.149521 - 01                                                                         -.514611 - 02                                                                         2.906  18.738                OHARA LAK02 6.499   1.669913                                                                           -.906677 - 01                                                                         -.139503 - 01                                                                         -.471304 - 02                                                                         2.960  19.238                SCHOTT                                                                              PSK53 6.508   1.620058                                                                           -.817978 - 01                                                                         -.125688 - 01                                                                         -.266930 - 02                                                                         4.709  30.644                OHARA BAK2  6.520   1.539892                                                                           -.872113 - 01                                                                         -.133754 - 01                                                                         -.336242 - 02                                                                         3.978  25.937                SCHOTT                                                                              BAK2  6.556   1.539893                                                                           -.870826 - 01                                                                         -.132837 - 01                                                                         -.374755 - 02                                                                         3.545  23.237                SCHOTT                                                                              SK12  6.568   1.583058                                                                           -.874398 - 01                                                                         -.133138 - 01                                                                         -.353850 - 02                                                                         3.763  24.711                HOYA  TAC1  6.590   1.725898                                                                           -.972090 - 01                                                                         -.147516 - 01                                                                         -.457236 - 02                                                                         3.226  21.260                SCHOTT                                                                              SK7   6.590   1.607214                                                                           -.874337 - 01                                                                         -.132673 - 01                                                                         -.337933 - 02                                                                         3.926  25.873                CORN-F                                                                              713538                                                                              6.606   1.712902                                                                           -.966678 - 01                                                                         -.146339 - 01                                                                         -.535544 - 02                                                                         2.733  18.050                OHARA KO1   6.610   1.523003                                                                           -.889419 - 01                                                                         -.134552 - 01                                                                         -.583532 - 02                                                                         2.306  15.242                HOYA  BAC2  6.622   1.539894                                                                           -.870947 - 01                                                                         -.131533 - 01                                                                         -.391689 - 02                                                                         3.358  22.236                HOYA  C3    6.658   1.518169                                                                           -.882289 - 01                                                                         -.132510 - 01                                                                         -.465668 - 02                                                                         2.846  18.947                HOYA  ZNC1  6.663   1.533081                                                                           -.894813 - 01                                                                         -.134302 - 01                                                                         -.328172 - 02                                                                         4.092  27.267                SCHOTT                                                                              K10   6.695   1.501305                                                                           -.921918 - 01                                                                         -.137707 - 01                                                                         -.724901 - 02                                                                         1.900  12.718                HOYA  C5    6.713   1.522423                                                                           -.875458 - 01                                                                         -.130419 - 01                                                                         -.233649 - 02                                                                         5.582  37.469                OHARA LAK010                                                                              6.714   1.725900                                                                           -.969928 - 01                                                                         -.144462 - 01                                                                         -.477007 - 02                                                                         3.029  20.334                OHARA K3    6.721   1.518164                                                                           -.882441 - 01                                                                         -.131296 - 01                                                                         -.392919 - 02                                                                         3.342  22.459                OHARA KZF5  6.796   1.521226                                                                           -.989376 - 01                                                                         -.145579 - 01                                                                         -.962169 - 02                                                                         1.513  10.283                SCHOTT                                                                              SK3   6.812   1.608733                                                                           -.881775 - 01                                                                         - .129447 - 01                                                                        -.355097 - 02                                                                         3.645  24.832                CHANCE                                                                              669574                                                                              6.830   1.668732                                                                           -.905610 - 01                                                                         -.132587 - 01                                                                         -.336905 - 02                                                                         3.935  26.880                HOYA  C12   6.844   1.523007                                                                           -.886949 - 01                                                                         -.129586 - 01                                                                         -.370803 - 02                                                                         3.495  23.920                SCHOTT                                                                              LAK23 6.851   1.668729                                                                           -.905576 - 01                                                                         -.132185 - 01                                                                         -.358112 - 02                                                                         3.691  25.287                HOYA  BACD7 6.855   1.607216                                                                           -.873371 - 01                                                                         -.127414 - 01                                                                         -.214385 - 02                                                                         5.943  40.738                OHARA K2    6.860   1.515949                                                                           -.914926 - 01                                                                         -.133366 - 01                                                                         -.549090 - 02                                                                         2.429  16.663                OHARA LAK8  6.895   1.712897                                                                           -.965187 - 01                                                                         -.139977 - 01                                                                         -.452127 - 02                                                                         3.096  21.346                SCHOTT                                                                              ZK1   6.901   1.533081                                                                           -.896179 - 01                                                                         -.129867 - 01                                                                         -.447048 - 02                                                                         2.905  20.047                CORN-F                                                                              573575                                                                              6.931   1.572425                                                                           -.904539 - 01                                                                         -.130502 - 01                                                                         -.351043 - 02                                                                         3.718  25.767                HOYA  LACL2 6.953   1.650109                                                                           -.934514 - 01                                                                         -.134408 - 01                                                                         -.589840 - 02                                                                         2.279  15.844                OHARA SK7   6.958   1.607216                                                                           -.874465 - 01                                                                         -.125676 - 01                                                                         -.262312 - 02                                                                         4.791  33.337                CORN-F                                                                              678555                                                                              6.978   1.677914                                                                           -.935998 - 01                                                                         -.134144 - 01                                                                         -.352755 - 02                                                                         3.803  26.534                CHANCE                                                                              658573                                                                              7.029   1.658213                                                                           -.907356 - 01                                                                         -.129095 - 01                                                                         -.329672 - 02                                                                         3.916  27.523                SCHOTT                                                                              PK51  7.035   1.528503                                                                           -.673937 - 01                                                                         -.957963 - 01                                                                         .453233 - 03                                                                          -21.136                                                                              -148.695              CHANCE                                                                              643580                                                                              7.044   1.642414                                                                           -.896155 - 01                                                                         -.127217 - 01                                                                         -.285256 - 02                                                                         4.460  31.416                OHARA SK3   7.093   1.608735                                                                           -.880631 - 01                                                                         -.124147 - 01                                                                         -.235977 - 02                                                                         5.261  37.319                SCHOTT                                                                              SK4   7.117   1.612643                                                                           -.885774 - 01                                                                         -.124460 - 01                                                                         -.283700 - 02                                                                         4.387  31.222                CORN-F                                                                              529518                                                                              7.163   1.528923                                                                           -.100340 + 00                                                                         -.140072 - 01                                                                         -.820824 - 02                                                                         1.706  12.224                CHANCE                                                                              572577                                                                              7.177   1.572126                                                                           -.899434 - 01                                                                         -.125326 - 01                                                                         -.381528 - 02                                                                         3.285  23.575                SCHOTT                                                                              LAKN6 7.183   1.642417                                                                           -.895749 - 01                                                                         - .124711 - 01                                                                        -.250159 - 02                                                                         4.985  35.807                OHARA BAK1  7.186   1.572428                                                                           -.899356 - 01                                                                         -.125151 - 01                                                                         -.366446 - 02                                                                         3.415  24.543                HOYA  BACD15                                                                              7.211   1.622915                                                                           -.892737 - 01                                                                         -.123807 - 01                                                                         -.308693 - 02                                                                         4.011  28.920                CHANCE                                                                              691547                                                                              7.229   1.690906                                                                           -.949247 - 01                                                                         -.131317 - 01                                                                         -.307996 - 02                                                                         4.264  30.820                SCHOTT                                                                              KZF2  7.233   1.529364                                                                           -.100562 + 00                                                                         -.139033 - 01                                                                         -.883979 - 02                                                                         1.573  11.376                OHARA ZK1   7.241   1.533083                                                                           -.895476 - 01                                                                         -.123663 - 01                                                                         -.319562 - 02                                                                         3.870  28.022                HOYA  BAC5  7.254   1.556638                                                                           -.883840 - 01                                                                         -.121843 - 01                                                                         -.324910 - 02                                                                         3.750  27.203                CHANCE                                                                              613586                                                                              7.301   1.612640                                                                           -.885524 - 01                                                                         -.121292 - 01                                                                         -.232589 - 02                                                                         5.215  38.073                OHARA LAK011                                                                              7.317   1.740893                                                                           -.986297 - 01                                                                         -.134798 - 01                                                                         -.417016 - 02                                                                         3.232  23.651                CORN-F                                                                              609590                                                                              7.373   1.608679                                                                           -.879735 - 01                                                                         -.119318 - 01                                                                         -.107720 - 02                                                                         11.077 81.669                OHARA KZF2  7.403   1.529366                                                                           -.100473 - 01                                                                         -.135712 - 01                                                                         -.873342 - 02                                                                         1.554  11.505                CHANCE                                                                              623581                                                                              7.415   1.622907                                                                           -.894283 - 01                                                                         -.120602 - 01                                                                         -.198530 - 02                                                                         6.075  45.045                HOYA  BACD4 7.444   1.612640                                                                           -.885957 - 01                                                                         -.119024 - 01                                                                         -.171424 - 02                                                                         6.943  51.682                HOYA  BACD3 7.459   1.608735                                                                           -.881444 - 01                                                                         -.118178 - 01                                                                         -.203789 - 02                                                                         5.799  43.253                OHARA LAK13 7.462   1.693404                                                                           -.972759 - 01                                                                         -.130362 - 01                                                                         -.437388 - 02                                                                         2.980  22.240                SCHOTT                                                                              SK15  7.532   1.622911                                                                           -.893772 - 01                                                                         -.118666 - 01                                                                         -.213001 - 02                                                                         5.571  41.961                HOYA  SBF6  7.598   1.526746                                                                           -.101704 + 00                                                                         -.133847 - 01                                                                         -.827299 - 02                                                                         1.618  12.293                SCHOTT                                                                              KZFS6 7.605   1.591875                                                                           -.107110 + 00                                                                         -.140833 - 01                                                                         -.115092 - 01                                                                         1.224  9.306                 SCHOTT                                                                              KZF6  7.606   1.526743                                                                           -.101593 + 00                                                                         -.133562 - 01                                                                         -.862104 - 02                                                                         1.549  11.784                OHARA PSK02 7.643   1.617927                                                                           -.818372 - 01                                                                         -.107073 - 01                                                                         -.110198 - 03                                                                         97.164 742.638               HOYA  SBF2  7.646   1.529364                                                                           -.100648 + 00                                                                         -.131629 - 01                                                                         -.780821 - 02                                                                         1.686  12.890                OHARA LAK11 7.657   1.658211                                                                           -.904552 - 01                                                                         -.118128 - 01                                                                         -.170177 - 02                                                                         6.941  53.154                SCHOTT                                                                              BAK5  7.677   1.556640                                                                           -.884378 - 01                                                                         -.115204 - 01                                                                         -.178067 - 02                                                                         6.470  49.665                CORN-F                                                                              651559                                                                              7.682   1.651038                                                                           -.928350 - 01                                                                         -.120842 - 01                                                                         -.210084 - 02                                                                         5.752  44.189                OHARA ZK4   7.685   1.511834                                                                           -.892013 - 01                                                                         -.116065 - 01                                                                         -.192179 - 02                                                                         6.039  46.416                SCHOTT                                                                              SK51  7.688   1.620824                                                                           -.859845 - 01                                                                         -.111842 - 02                                                                         -.188495 -02                                                                          5.933  45.616                HOYA  C10   7.703   1.501305                                                                           -.923216 - 01                                                                         -.119846 - 01                                                                         -.567892 - 02                                                                         2.110  16.257                HOYA  TAC2  7.741   1.740893                                                                           -.984759 - 01                                                                         -.127209 - 01                                                                         -.236823 - 02                                                                         5.371  41.582                OHARA LAK013                                                                              7.747   1.640915                                                                           -.911302 - 01                                                                         -.117626 - 01                                                                         -317403 - 02                                                                          3.706  28.711                OHARA SK15  7.779   1.622912                                                                           -.892144 - 01                                                                         -.114685 - 01                                                                         -.154793 - 02                                                                         7.409  57.635                CORN-F                                                                              691548                                                                              7.823   1.690915                                                                           -.947209 - 01                                                                         -.121087 - 01                                                                         -.207882 - 02                                                                         5.825  45.565                OHARA SK4   7.852   1.612638                                                                           -.882629 - 01                                                                         -.112411 - 01                                                                         -.811949 - 03                                                                         13.844 108.705               SCHOTT                                                                              SK52  7.868   1.638455                                                                           -.934108 - 01                                                                         -.118722 - 01                                                                         -.268132 - 02                                                                         4.428  34.838                CORN-F                                                                              658572                                                                              7.983   1.657418                                                                           -.906808 - 01                                                                         -.113589 - 01                                                                         -.118132 - 02                                                                         9.615  76.763                OHARA KZF6  8.073   1.526745                                                                           -.101431 + 00                                                                         -.125642 - 01                                                                         -.806489 - 02                                                                         1.558  12.577                HOYA  BACD13                                                                              8.081   1.591735                                                                           -.889125 - 01                                                                         -.110026 - 01                                                                         -.112692 - 02                                                                         9.763  78.898                OHARA LASK01                                                                              8.106   1.754892                                                                           -.991531 - 01                                                                         -.122321 - 01                                                                         -.261430 - 02                                                                         4.679  37.927                SCHOTT                                                                              SK13  8.157   1.591734                                                                           -.888920 - 01                                                                         -.108980 - 01                                                                         -.861306 - 03                                                                         12.653 103.206               OHARA LAK09 8.164   1.733892                                                                           -.100890 + 00                                                                         -.123579 -  01                                                                        -.374200 - 02                                                                         3.302  26.961                SCHOTT                                                                              LAKN16                                                                              8.186   1.733395                                                                           -.100420 + 00                                                                         -.122675 - 01                                                                         -.353129 - 02                                                                         3.474  28.437                HOYA  BAC4  8.396   1.568754                                                                           -.925165 - 01                                                                         -.110188 - 01                                                                         -.271226 - 02                                                                         4.063  34.110                OHARA BAK5  8.442   1.556642                                                                           -.883742 - 01                                                                         -.104688 - 01                                                                         .113065 - 04                                                                          -925.906                                                                             -7816.220             CORN-F                                                                              604640                                                                              8.470   1.603439                                                                           -.809551 - 01                                                                         -.955736 - 02                                                                         .201133 - 02                                                                          -4.752 -40.249               SCHOTT                                                                              BAK6  8.480   1.574364                                                                           -.918813 - 01                                                                         -.108344 - 01                                                                         -.311720 - 02                                                                         3.476  29.476                OHARA SK19  8.485   1.613341                                                                           -.902771 - 01                                                                         -.106402 - 01                                                                         -.113363 - 02                                                                         9.386  79.635                HOYA  LAC12 8.544   1.677812                                                                           -.932141 - 01                                                                         -.109101 - 01                                                                         -.446700 - 03                                                                         24.424 208.673               OHARA SK10  8.731   1.622718                                                                           -.907040 - 01                                                                         -.103893 - 01                                                                         -.105581 - 02                                                                         9.840  85.909                SCHOTT                                                                              KF3   8.780   1.514471                                                                           -.947050 - 01                                                                         -.107861 - 01                                                                         -.380584 - 02                                                                         2.834  24.884                SCHOTT                                                                              BAK4  8.842   1.568755                                                                           -.922830 - 01                                                                         -.104366 - 01                                                                         -.200230 - 02                                                                         5.212  46.089                CHANCE                                                                              569561                                                                              8.976   1.568752                                                                           -.922182 - 01                                                                         -.102743 - 01                                                                         -.154966 - 02                                                                         6.630  59.509                OHARA LAK12 8.976   1.677809                                                                           -.935008 - 01                                                                         -.104171 - 01                                                                         -.405546 - 03                                                                         25.687 230.555               SCHOTT                                                                              SK19  9.015   1.613342                                                                           -.902379 - 01                                                                         -.100103 - 01                                                                         -.251304 - 03                                                                         39.833 359.078               SCHOTT                                                                              LAK10 9.053   1.719894                                                                           -.102773 + 00                                                                         -.113521 - 01                                                                         -.488559 - 02                                                                         2.324  21.036                HOYA  BAC6  9.077   1.574366                                                                           -.919126 - 01                                                                         -.101259 - 01                                                                         -.220226 - 02                                                                         4.598  41.735                SCHOTT                                                                              LAKN22                                                                              9.114   1.651044                                                                           -.926400 - 01                                                                         -.101646 - 01                                                                         .130301 - 03                                                                          -78.009                                                                              -710.970              OHARA LAK04 9.118   1.650910                                                                           -.921923 - 01                                                                         -.101114 - 01                                                                         -.135790 - 02                                                                         7.446  67.894                HOYA  CF3   9.168   1.514470                                                                           -.948556 - 01                                                                         -.103464 - 01                                                                         -.392678 - 02                                                                         2.635  24.156                OHARA K10   9.179   1.501306                                                                           -.918246 - 01                                                                         -.100037 - 01                                                                         -.722028 -02                                                                          1.385  12.718                SCHOTT                                                                              KZFSN9                                                                              9.220   1.598463                                                                           -.110530 + 00                                                                         -.119880 - 01                                                                         -.993952 - 02                                                                         1.206  11.120                OHARA LAF03 9.263   1.734887                                                                           -.103971 + 00                                                                         -.112248 - 01                                                                         -.398197 - 02                                                                         2.819  26.111                CORN-F                                                                              569560                                                                              9.314   1.568722                                                                           -.924370 - 01                                                                         -.992437 - 02                                                                         -.959931 - 03                                                                         10.339 96.295                SCHOTT                                                                              LAK28 9.335   1.744179                                                                           -.101990 + 00                                                                         -.109256 - 01                                                                         -.222904 - 02                                                                         4.901  45.755                SCHOTT                                                                              BAK1  9.337   1.572426                                                                           -.899074 - 01                                                                         -.962889 - 02                                                                         .348458 - 03                                                                          -27.633                                                                              -258.015              OHARA BAK4  9.365   1.568756                                                                           -.921768 - 01                                                                         -.984275 - 02                                                                         -.991787 - 03                                                                         9.924  92.940                OHARA BAK6  9.392   1.574366                                                                           -.916184 - 01                                                                         -.975546 - 02                                                                         -.244339 - 03                                                                         39.926 374.964               HOYA  TAC4  9.414   1.733893                                                                           -.101592 + 00                                                                         -.107914 - 01                                                                         .141620 - 03                                                                          -76.200                                                                              -717.353              HOYA  BAC1  9.426   1.572424                                                                           -.901217 - 01                                                                         -.956143 - 02                                                                         -.168226 - 03                                                                         56.837 535.719               SCHOTT                                                                              SKN18 9.569   1.638456                                                                           -.933396 - 01                                                                         -.975467 - 02                                                                         -.181507 - 02                                                                         5.374  51.425                SCHOTT                                                                              SK10  9.598   1.622720                                                                           -.908955 - 01                                                                         -.947032 - 02                                                                         .369531 - 03                                                                          -25.628                                                                              -245.976              OHARA KF3   9.645   1.514470                                                                           -.946095 - 01                                                                         -.980925 - 02                                                                         -.134816 - 02                                                                         7.276  70.177                CHANCE                                                                              623569                                                                              9.672   1.622716                                                                           -.909436 - 01                                                                         -.940274 - 02                                                                         .958374 - 03                                                                          -9.811 -94.894               CORN-F                                                                              734514                                                                              9.796   1.733407                                                                           -.100712 + 00                                                                         -.102813 - 01                                                                         -.543492 - 03                                                                         18.917 185.305               CHANCE                                                                              678552                                                                              9.798   1.677809                                                                           -.937135 - 01                                                                         -.956504 - 02                                                                         .811637 - 03                                                                          -11.785                                                                              -115.462              OHARA SK18  9.850   1.638454                                                                           -.933611 - 01                                                                         -.947800 - 02                                                                         -.242891 - 02                                                                         3.902  38.437                SCHOTT                                                                              SK1   9.906   1.610171                                                                           -.912011 - 01                                                                         -.920641 - 02                                                                         .728678 - 03                                                                          -12.634                                                                              -125.160              CHANCE                                                                              734517                                                                              10.006  1.733395                                                                           -.100140 + 00                                                                         -.100080 - 01                                                                         .441798 - 04                                                                          -226.528                                                                             -2266.650             HOYA  BACD10                                                                              10.054  1.622719                                                                           -.908052 - 01                                                                         -.903132 - 02                                                                         .146881 -02                                                                           -6.149 -61.822               CORN-F                                                                              548535                                                                              10.127  1.547678                                                                           -.968188 - 01                                                                         -.956016 - 02                                                                         -.428149 - 02                                                                         2.233  22.613                CORN-F                                                                              623569                                                                              10.167  1.622720                                                                           -.908938 - 01                                                                         -.894014 - 02                                                                         .186653 - 02                                                                          -4.790 -48.697               HOYA  BACD1 10.204  1.610168                                                                           -.912395 - 01                                                                         -.894152 - 02                                                                         .160556 - 02                                                                          -5.569 -56.827               SCHOTT                                                                              SK2   10.234  1.607302                                                                           -.912395 - 01                                                                         -.891538 - 02                                                                         .132736 - 02                                                                          -6.717 -68.737               CHANCE                                                                              639554                                                                              10.318  1.638455                                                                           -.932891 - 01                                                                         -.904120 - 02                                                                         -.105759 - 02                                                                         8.549  88.209                HOYA  BACD18                                                                              10.325  1.638456                                                                           -.932041 - 01                                                                         -.902681 - 02                                                                         -.515061 - 03                                                                         17.526 180.957               SCHOTT                                                                              KZFS1 10.361  1.612997                                                                           -.116813 + 00                                                                         -.112746 - 01                                                                         -.107927 - 01                                                                         1.045  10.823                HOYA  BACD2 10.377  1.607305                                                                           -.912003 - 01                                                                         -.878874 - 02                                                                         .199064 - 02                                                                          -4.415 -45.815               CORN-F                                                                              614564                                                                              10.505  1.613418                                                                           -.916306 - 01                                                                         -.872255 -  20180 - 02                                                                -.270   -51.976                      HOYA  TAFD1 205.243 1.814564                                                                           -.115964 + 00                                                                         -.565007 - 03                                                                         .622681 - 02                                                                          -.091  -18.623               HOYA  NBFD14                                                                              224.677 1.801566                                                                           -.115564 + 00                                                                         -.514356 - 03                                                                         .564119 - 02                                                                          -.091  -20.486               HOYA  TAFD10                                                                              258.348 1.815368                                                                           -.115241 + 00                                                                         -.446069 - 03                                                                         .763394 - 02                                                                          -.058  -15.096               OHARA LLF2  363.647 1.540638                                                                           -.108587 + 00                                                                         -.298606 - 03                                                                         .161587 - 02                                                                          -.185  -67.200               SCHOTT                                                                              LGSK2 452.288 1.585919                                                                           -.837990 - 01                                                                         -.185278 - 03                                                                         .107401 - 01                                                                          -.017  -7.802                OHARA LAFO6 978.209 1.685896                                                                           -.104144 + 00                                                                         -.106464 - 03                                                                         .570256 - 02                                                                          -.019  -18.263               CORN-F                                                                              655401                                                                              1271.950                                                                              1.655380                                                                           -.127954 + 00                                                                         -.100597 - 03                                                                         -.301127 - 02                                                                         .033   42.492                __________________________________________________________________________

I claim:
 1. An optical system rotationally symmetric about an axis, saidsystem comprising a lens quadruplet having a common focal length on saidaxis for four optical wavelengths, said lens quadruplet comprising afirst lens element made of an optical material having a refractive indexof approximately 1.49 and an Abbe number of approximately 81.40 at abase wavelength of 0.546 micrometers, a second lens element made of anoptical material having a refractive index of approximately 1.59 and anAbbe number of approximately 60.94 at said base wavelength, a third lenselement and a fourth lens element, one of said third and fourth lenselements being made of the same optical material as said first lenselement, and another one of said third and fourth lens elements beingmade of the same optical material as said second lens element.
 2. Theoptical system of claim 1 wherein said third lens element is made of thesame optical material as said first lens element, said optical systemhaving a design form wherein said lens elements are positioned so thatsaid first and second lens elements are in contact with each other, saidsecond and third lens elements are in contact with each other, and saidthird and fourth lens elements are in contact with each other, therebydefining first, second, third, fourth and fifth surfaces, said firstlens element being bounded by said first and second surfaces, saidsecond lens element being bounded by said second and third surfaces,said third lens element being bounded by said third and fourth surfaces,said fourth lens element being bounded by said fourth and fifthsurfaces, said first surface having a positive radius of curvature ofapproximately 0.437 times the focal length of said quadruplet, saidsecond surface having a positive radius of curvature of approximately5.098 times said focal length, said third surface having a positiveradius of curvature of 0.285 times said focal length, said fourthsurface having a negative radius of curvature of approximately 0.363times said focal length, said fifth surface having a negative radius ofcurvature of approximately 1.147 times said focal length, said firstlens element having a thickness along said axis of approximately 0.015times said focal length, said second lens element having a thicknessalong said axis of approximately 0.009 times said focal length, saidthird lens element having a thickness along said axis of approximately0.015 times said focal length, said fourth lens element having athickness along said axis of approximately 0.009 times said focallength.
 3. The optical system of claim 1 wherein said third lens elementis made of the same optical material as said second lens element, saidoptical system having a design form wherein said lens elements arepositioned so that said first and second lens elements are in contactwith each other, and so that said third and fourth lens elements are incontact with each other, thereby defining first, second, third, fourth,fifth and sixth surfaces, said first lens element being bounded by saidfirst and second surfaces, said second lens element being bounded bysaid second and third surfaces, said third element being bounded by saidfourth and fifth surfaces, said fourth lens element being bounded bysaid fifth and sixth surfaces, said first surface having a positiveradius of curvature of approximately 0.948 times the focal length ofsaid quadruplet, said second surface having a negative radius ofcurvature of approximately 0.375 times said focal length, said thirdsurface having a positive radius of curvature of approximately 1.280times said focal length, said fourth surface having a positive radius ofcurvature of approximately 0.499 times said focal length, said fifthsurface having a positive radius of curvature of approximately 0.274times said focal length, said sixth surface having a negative radius ofcurvature of approximately 1.150 times said focal length, said firstlens element having a thickness along said axis of approximately 0.015times said focal length, said second lens element having a thicknessalong said axis of approximately 0.009 times said focal length, saidthird lens element being separated from said second lens element alongsaid axis by approximately 0.001 times said focal length, said thirdlens element having a thickness along said axis of approximately 0.009times said focal length, said fourth lens element having a thicknessalong said axis of approximately 0.015 times said focal length.
 4. Theoptical system of claim 1 wherein said third lens element is made of thesame optical material as said first lens element, said optical systemhaving a design form wherein said lens elements are positioned so thatsaid first and second lens elements are in contact with each other, andso that said third and fourth lens elements are in contact with eachother, thereby defining first, second, third, fourth, fifth and sixthsurfaces, said first lens element being bounded by said first and secondsurfaces, said second lens element being bounded by said second andthird surfaces, said third lens elements being bounded by said fourthand fifth surfaces, said fourth lens elements being bounded by saidfifth and sixth surfaces, said first surface having a positive radius ofcurvature of approximately 0.823 times the focal length of saidquadruplet, said second surface having a negative radius of curvature ofapproximately 0.400 times said focal length, said third surface having apositive radius of curvature of approximately 1.069 times said focallength, said fourth surface having a positive radius of curvature ofapproximately 0.645 times said focal length, said fifth surface having anegative radius of curvature of approximately 0.337 times said focallength, said sixth surface having a negative radius of curvature ofapproximately 0.758 times said focal length, said first lens elementhaving a thickness along said axis of approximately 0.015 times saidfocal length, said second lens element having a thickness along saidaxis of approximately 0.009 times said focal length, said third lenselement being separated from said second lens element along said axis byapproximately 0.001 times said focal length, said third lens elementhaving a thickness along said axis of approximately 0.015 times saidfocal length, said fourth lens element having a thickness along saidaxis of approximately 0.009 times said focal length.
 5. An opticalsystem for collimating more than two optical wavelengths, said systemcomprising a plurality of refractive elements disposed along an opticaxis of said system, said refractive elements being made from only twodifferent optical materials, at least one of said refractive elementsbeing made of a first one of said optical materials and at least oneother of said refractive elements being made of a second one of saidoptical materials, said refractive elements coacting with each other sothat more than two wavelengths in a beam of optical radiation passingthrough said system are precisely collimated with respect to said opticaxis.
 6. The optical system of claim 5, said system being a lensmultiplet for collimating four optical wavelengths.
 7. The opticalsystem of claim 6 wherein said lens multiplet comprises a lensquadruplet having a common focal length along an axis for said fouroptical wavelengths, said lens quadruplet comprising a first lenselement made of an optical material having a refractive index ofapproximately 1.46 and an Abbe number of approximately 90.32 at a basewavelength of 0.546 micrometers, a second lens element made of anoptical material having a refractive index of approximately 1.70 and anAbbe number of approximately 55.31 at said base wavelength, a third lenselement made of the same optical material as said second lens element,and a fourth lens element made of the same optical material as saidfirst lens element.
 8. The optical system of claim 7 having a designform wherein said lens elements are positioned with respect to eachother to define first, second, third, fourth, fifth, sixth, seventh andeighth surfaces, said first lens element being bounded by said first andsecond surfaces, said second lens element being bounded by said thirdand fourth surfaces, said third lens element being bounded by said fifthand sixth surfaces, said fourth lens element being bounded by saidseventh and eighth surfaces, said first surface having a positive radiusof curvature of approximately 1.244 times the focal length of saidquadruplet, said second surfaces having a negative radius of curvatureof approximately 0.381 times said focal length, said third surfacehaving a negative radius of curvature of approximately 0.381 times saidfocal length, said fourth surface having a negative radius of curvatureof approximately 0.542 times said focal length, said fifth surfacehaving a negative radius of curvature of approximately 0.539 times saidfocal length, said sixth surface having a negative radius of curvatureof approximately 3.845 times said focal length, said seventh surfacehaving a positive radius of curvature of approximately 0.715 times saidfocal length, said eighth surface having a negative radius of curvatureof approximately 1.025 times said focal length, said first lens elementhaving a thickness along said axis of approximately 0.015 times saidfocal length, said second lens element being separated from said firstlens element along said axis by approximately 0.010 times said focallength, said second lens element having a thickness along said axis ofapproximately 0.009 times said focal length, said third lens elementbeing separated from said second lens element along said axis byapproximately 0.002 times said focal length, said third lens elementhaving a thickness along said axis of approximately 0.009 times saidfocal length, said fourth lens element being separated from said thirdlens element along said axis by approximately 0.0005 times said focallength, said fourth lens element having a thickness along said axis ofapproximately 0.015 times said focal length.
 9. The optical system ofclaim 5, said system comprising a lens multiplet for collimating fiveoptical wavelengths.
 10. The optical system of claim 9 wherein said lensmultiplet comprises a lens quadruplet having a common focal length alongsaid axis for said five optical wavelengths, said lens quadrupletcomprising a first lens element made of an optical material having arefractive index of approximately 1.53 and an Abbe number ofapproximately 76.57 at a base wavelength of 0.546 micrometers, a secondlens element made of an optical material having a refractive index ofapproximately 1.65 and an Abbe number of approximately 58.15 at saidbase wavelength, a third lens element made of the same optical materialas said second lens element, and a fourth lens element made of the sameoptical material as said first lens element.
 11. The optical system ofclaim 10 having a design form wherein said lens elements are positionedwith respect to each other to define first, second, third, fourth,fifth, sixth, seventh and eighth surfaces, said first lens element beingbounded by said first and second surfaces, said second lens elementbeing bounded by said third and fourth surfaces, said third lens elementbeing bounded by said fifth and sixth surfaces, said fourth lens elementbeing bounded by said seventh and eighth surfaces, said first surfacehaving a negative radius of curvature of approximately 1.314 times thefocal length of said quadruplet, said second surface having a negativeradius of curvature of approximately 0.295 times said focal length, saidthird surface having a negative radius of curvature of approximately0.269 times said focal length, said fourth surface having a negativeradius of curvature of approximately 1.795 times said focal length, saidfifth surface having a positive radius of curvature of approximately0.469 times said focal length, said sixth surface having a positiveradius of curvature of approximately 0.233 times said focal length, saidseventh surface having a positive radius of curvature of approximately0.232 times said focal length, said eighth surface having a negativeradius of curvature of approximately 0.697 times said focal length, saidfirst lens element having a thickness along said axis of approximately0.018 times said focal length, said second lens element being separatedfrom said first lens element along said axis by approximately 0.025times said focal length, said second lens element having a thicknessalong said axis of approximately 0.010 times said focal length, saidthird lens element being separated from said second lens element alongsaid axis by approximately 0.003 times said focal length, said thirdlens element having a thickness having a thickness along said axis ofapproximately 0.010 times said focal length, said fourth lens elementbeing separated from said third lens element along said axis byapproximately 0.002 times said focal length, said fourth lens elementhaving a thickness along said axis of approximately 0.015 times saidfocal length.
 12. The optical system of claim 5 wherein one of saidoptical materials comprises a crystal.
 13. The optical system of claim12, said system being a lens multiplet for collimating five opticalwavelengths.
 14. The optical system of claim 13 wherein said lensmultiplet comprises a lens quadruplet having a common focal length alongsaid axis for said five optical wavelengths, said lens quadrupletcomprising a first lens element made of calcium fluoride, a second lenselement made of an optical material having a refractive index ofapproximately 1.65 and an Abbe number of approximately 58.15 at a basewavelength of 0.546 micrometers, a third lens element made of the sameoptical material as said second lens element, and a fourth lens elementmade of calcium fluoride.
 15. The optical system of claim 14 having adesign form wherein said lens elements are positioned with respect toeach other to define first, second, third, fourth, fifth, sixth, seventhand eighth surfaces, said first lens element being bounded by said firstand second surfaces, said second lens element being bounded by saidthird and fourth surfaces, said third lens element being bounded by saidfifth and sixth surfaces, said fourth lens element being bounded by saidseventh and eighth surfaces, said first surface having a negative radiusof curvature of approximately 0.668 times the focal length of saidquadruplet, said second surface having a negative radius of curvature ofapproximately 0.268 times said focal length, said third surface having anegative radius of curvature of approximately 0.250 times said focallength, said fourth surface having a negative radius of curvature ofapproximately 0.565 times said focal length, said fifth surface having apositive radius of curvature of approximately 0.455 times said focallength, said sixth surface having a positive radius of curvature ofapproximately 0.384 times said focal length, said seventh surface havinga positive radius of curvature of approximately 0.424 times said focallength, said eighth surface having a negative radius of curvature ofapproximately 0.635 times said focal length, said first lens elementhaving a thickness along said axis of approximately 0.018 times saidfocal length, said second lens element being separated from said firstlens element along said axis by approximately 0.032 times said focallength, said second lens element having a thickness along said axis ofapproximately 0.010 times said focal length, said third lens elementbeing separated from said second lens element along said axis byapproximately 0.001 times said focal length, said third lens elementhaving a thickness along said axis of approximately 0.010 times saidfocal length, said fourth lens element being separated from said thirdlens element along said axis by approximately 0.001 times said focallength, said fourth lens element being separated from said third lenselement along said axis by approximately 0.001 times said focal length,said fourth lens element having a thickness along said axis ofapproximately 0.015 times said focal length.
 16. A dioptric opticalsystem consisting of a lens quadruplet, said lens quadruplet being madefrom only two different optical materials, namely, Corning-France 486817glass and Corning-France 589612 glass, at least one lens element of saidlens quadruplet being made of Corning-France 486817 glass and at leastone other lens element of said lens quadruplet being made ofCorning-France 589612 glass, said lens quadruplet being color-correctedso that paraxial marginal rays passing through said lens quadruplet cometo a common focus at four discrete wavelengths.
 17. The optical systemof claim 16, said lens quadruplet comprising four lens elements, two ofsaid lens elements being air-spaced.
 18. A dioptric optical systemprecisely color-corrected at exactly four discrete wavelengths, saidsystem consisting of a plurality of refractive elements, said refractiveelements being made from only two different optical materials, namely,Hoya FCD10 glass and Ohara LAK14 glass, at least one of said refractiveelements being made of Hoya FCD10 glass and at least one other of saidrefractive elements being made of Ohara LAK14 glass, said refractiveelements coacting with each other so as to achieve precise colorcorrection at said four discrete wavelengths for said system.
 19. Theoptical system of claim 18, said system being being a collimator. 20.The optical system of claim 19, said collimator comprising four lenselements.
 21. The optical system of claim 20, said lens elements beingair-spaced.
 22. A dioptric optical system precisely color-corrected atexactly five wavelengths, said system consisting of a plurality ofrefractive elements, said refractive elements being made from only twodifferent optical materials, namely, Schott PK51 glass and Hoya LAC7glass, at least one of said refractive elements being made of SchottPK51 glass and at least one other of said refractive elements being madeof Hoya LAC7 glass, said refractive elements coacting with each other soas to achieve precise color correction at said five discrete wavelengthsfor said system.
 23. The optical system of claim 22, said system being acollimator.
 24. The optical system of claim 23, said collimatorcomprising four lens elements.
 25. The optical system of claim 24, saidlens elements being air-spaced.
 26. A dioptric optical system preciselycolor-corrected at exactly five discrete wavelengths, said systemconsisting of a plurality of refractive elements, said refractiveelements being made from only two different optical materials, namely,calcium fluoride crystal and Hoya LAC7 glass, at least one of saidrefractive elements being made of calcium fluoride crystal and at leastone other of said refractive elements being made of Hoya LAC7 glass,said refractive elements coacting with each other so as to achieveprecise color correction at said five discrete wavelengths for saidsystem.
 27. The optical system of claim 26, said system being acollimator.
 28. The optical system of claim 27, said collimatorcomprising four lens elements.
 29. The optical system of claim 28, saidlens elements being air-spaced.
 30. A dioptric optical system comprisingfour lens elements positioned along an optic axis, each of said lenselements being disposed symmetrically about said axis, two of said lenselements being made of a first optical material and another two of saidlens elements being made of a second optical material, each lens elementmade of said first optical material being of positive dioptric power,each lens element made of said second optical material being of negativedioptric power, said four lens elements coacting with each other tobring paraxial marginal rays to a common focus on said axis at more thantwo discrete wavelengths.
 31. The dioptric optical system of claim 30wherein said first optical material is Corning-France 486817 glass andsaid second optical material is Corning-France 589612 glass.
 32. Thedioptric optical system of claim 31 wherein said four lens elementscoact to bring paraxial marginal rays to said common focus at fourdiscrete wavelengths.
 33. The dioptric optical system of claim 30wherein said first optical material is Hoya FCD10 glass and said secondoptical material is Ohara LAK14 glass.
 34. The dioptric optical systemof claim 33 wherein said four lens elements coact to bring paraxialmarginal rays to said common focus at four discrete wavelengths.
 35. Thedioptric optical system of claim 30 wherein said first optical materialis Schott PK51 glass and said second optical material is Hoya LAC7glass.
 36. The dioptric optical system of claim 35 wherein said fourlens elements coact to bring paraxial marginal rays to said common focusat five discrete wavelengths.
 37. The dioptric optical system of claim30 wherein said first optical material is calcium fluoride crystal andsaid second optical material is a glass.
 38. The dioptric optical systemof claim 37 wherein said four lens elements coact to bring paraxialmarginal rays to said common focus at five discrete wavelengths.
 39. Thedioptric optical system of claim 37 wherein said second optical materialis Hoya LAC7 glass.
 40. A dioptric optical system comprising a pluralityof lens elements positioned along an optic axis, each of said lenselements being disposed symmetrically about said axis, each of said lenselements consisting of one of only two different optical materials, atleast one of said lens elements being made from a first one of saidoptical materials, at least one other of said lens elements being madefrom a second one of said optical materials, all lens elements that aremade from said first one of said optical materials being of positivedioptric power, all lens elements that are made from said second one ofsaid optical materials being of negative dioptric power, said pluralityof lens elements coacting with each other to bring paraxial marginalrays to a common focus on said axis at more than two discretewavelengths.
 41. The dioptric optical system of claim 40 wherein saidplurality of lens elements comprises four lens elements, two of saidlens elements being made from said first one of said optical materials,another two of said lens elements being made from said second one ofsaid optical materials.
 42. The dioptric optical system of claim 41wherein said four lens elements coact to bring paraxial marginal rays tosaid common focus at four discrete wavelengths.
 43. The dioptric opticalsystem of claim 41 wherein said four lens elements coact to bringparaxial marginal ray to said common focus at five discrete wavelengths.